MAT125 AllofUnit2

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53 Section I Linear Regression So far we have discussed describing and summarizing one variable, but very often we want to know if two or more variables are related and if they are related, we want to describe that relationship. One way to analyze the relationship between two or more variables is a method called linear regression, specifically least squares linear regression. In this section, we will only look at the relationship between two variables. Note: least squares linear regression is not the only type of regression analysis, but it is the only type discussed in this course. An ordered pair consists of values of two variables for each individual in the data set. Data that consist of ordered pairs is called bivariate data. Response variable (Dependent variable) is the variable whose value can be explained by the value of the explanatory or predicator variable (independent variable) . Example: GPA depends on Number of Hours Studied, Height depends on Shoe Size Scatter plot is a graph that shows the relationship between two quantitative variables, measured on the same individual. Explanatory variable is on the horizontal axis (x-axis) Response variable is on the vertical axis (y-axis) Note: cannot always be sure which is which (does weight depend on height? or does height depend on weight?) Determine whether a linear, nonlinear or no relationship exists: We don’t just want to look at a scatter plot to determine if there is a relationship we also want to determine how strong that linear relationship is, therefore we want to find the linear correlation coefficient .

54 The linear correlation coefficient is a measure of the strength and direction of the linear relation between two quantitative variables. We use the Greek letter ρ (rho) to represent the population correlation coefficient and r to represent the sample correlation coefficient. The following is the formula for the sample correlation coefficient: r = ∑ xy − ∑ x ∑ y n √ (∑ x 2 − (∑ x) 2 n )√(∑ ? 2 − (∑y) 2 n ) Please note: you will NOT be using this formula to calculate the linear correlation coefficient, you will be learning how to use your calculator and reading a Minitab printout to find the linear correlation coefficient. Properties of the Linear Correlation Coefficient 1) − 1 ≤ r ≤ 1 2) If r = +1, then a perfect positive linear relation exists between the two variables. 3) If r = − 1, then a perfect negative linear relation exists between the two variables. 4) The closer r is to +1, the stronger is the evidence of positive linear association between the two variables. 5) The closer r is to −1, the stronger is the evidence of negative linear association between the two variables. 6) If r is close to 0, then little or no evidence exists of a linear relation between the two variables. Note: the linear correlation coefficient is a measure of the strength of the linear relation, r close to 0 does not imply no relation, just no linear relation. 7) The linear correlation coefficient is a unitless measure of association. 8) The correlation coefficient is not resistant. Therefore, an observation that does not follow the overall pattern of the data could affect the value of the linear correlation coefficient. Note: Correlation is not the same as causation. In general, when two variables are correlated we cannot conclude that changing the value of one variable will cause a change in the value of the other. In other words, correlation shows that two variables are associated, but not that one actually causes the other to occur. They both may be caused by a third variable. Least-Squares Regression Now we know that two variables have a linear relation, we want to find a line that best fits the points. One way to do this is to pick two points that appear to be a good fit of the data and find the line through those points. But is this the best line? Meaning will the predictions made be accurate. The method we will be using to find the line that best fits the data is called least-squares regression . The line found by using this method is the line in which the sum of the squared vertical distances from the observed value and the line is as small as possible. This line is called the least-squares regression line. The least-squares regression line is written as a linear equation containing two variables, x and y ̂ and an equal sign.

55 Least-Squares Linear Regression Model The sum of these distances squared would be the smallest for all the lines draw to fit these points. Finding the Least-Squares Regression Line Given ordered pairs ( x, y ), with means x and y , sample standard deviations s x and s y , and correlation coefficient r , the equation of the least-squares regression line for predicting y from x is y ̂ = b o + b 1 x where b 1 = r ∗ s y s x is the slope and b o = y ̅ − b 1 x ̅ is the y-intercept In general, the variable we want to predict is call the response variable (dependent variable) and the variable we are given is called the explanatory variable or predictor variable (independent variable). Please note: you will NOT be using these formulas to calculate the slope or y-intercept, you will be learning how to use your calculator and reading a Minitab printout to find the slope and y-intercept. Note: The least-squares regression line goes through the point of averages ) , ( y x . Note: If r is positive, then the slope is positive. If r is negative, then the slope is negative. Interpretation of Slope: (change in y)/(change in x) The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average. Example: For a line whose slope is 1.35, if x increases by 1, y will increase by 1.35. If a line whose slope is – 3, if x increases by 1, y will decrease by 3. Interpretation of y-intercept: If the y-intercept is near the observed values, then the y-intercept is the value of the predicted y value when the x value is zero. If the y-intercept is not near the observed values then the y-intercept does not have a useful interpretation. Diagnostics on the Least-Squares Regression Line You don’t want to use the least squares regression line to make predictions of the explanatory variable (x- values) that are much larger or much smaller than those observed. We don’t know what happens outside the scope of the observed values, therefore you should not use the regression model to make predictions outside the scope of the model. Making predictions for values for the explanatory variable that are outside the range of the data is called extrapolation.

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56 Residual Analysis is used to determine whether a linear model is appropriate to describe the relation between the explanatory and response variables. Given a point (x, y) on a scatterplot, and the least-squares regression line y ̂ = b o + b 1 x , the residual for the point (x, y) is the difference between the observed value of y and the predicted value y ̂ . Residual = error = y − ? ̂ Least-Squares Linear Regression Model predicted value, y ̂ i = b o + b 1 x Residual = error = y i − y ̂ i observed value, y i For example, if the least squares regression equation is found to be y ̂ = 10 + 6x and one of the observed points from the data set was (3, 25.75). Then the predicted value when x = 3 would be y ̂ = 10 + 6(3) = 28 . So the residual for this observation would be, residual = 25.75 – 28 = − 2.25 . Note: The residuals are positive for points above the line and negative for points below the line. The least-squares regression line satisfies the least-squares property. This means that the sum of the squared residuals is less for the least-squares regression line than for any other line. A residual plot is a plot in which the residuals are plotted against the values of the explanatory variable x. Note: 1)When a residual plot exhibits a noticeable pattern, the variables do not have a linear relationship, and the least-squares regression line should not be used. 2) When a residual plot exhibits no noticeable pattern, the least-squares line may be used to describe the relationship between the variables.

57 In which plot below would a least-squares line be used to describe the relationship between two variables? Why? A) B) You cannot just rely on the correlation coefficient to determine whether two variables have a linear relationship; e ven when the correlation is close to 1 or −1, the rel ationship may not be linear, a residual should be constructed to determine whether two variables have a linear relationship. Determining Outliers and Influential Points in a Regression Model An outlier is an observation that does not fit the overall pattern of the data. An outlier can be determined by a residual plot, a boxplot of the residuals or using a Minitab printout. An outlier has a standard residual that is either greater than 2 or less than -2. An influential point is a point that, when included in a scatterplot, strongly affects the position of the least-squares regression line. i.e. an influential point is an observation that significantly affects the value of the slope and/or y-intercept of the least-squares regression line and the value of the correlation coefficient. Please note you will using a Minitab printout to determine outliers and/or influential observations. Coefficient of Determination, r 2 , measures the proportion of total variation in the response variable that is explained by the least-squares regression line. Since r 2 is a proportion, it can never be negative or greater than 1. ( 0 ≤ r 2 ≤ 1 ) r 2 = 0 means the least-squares regression line has no explanatory value r 2 = 1 means the least-squares regression line explains 100% of the variation in the response variable (i.e. the closer r 2 is to 1, the closer the predictions made by the least-squares regression line are to the actual values, on average.) The coefficient of determination is a measure of how well the least-square regression line describes the relation between the explanatory and response variable. The closer r 2 is to 1, the better the line describes how changes in the explanatory variable affect the value of the response variable. Note: Squaring the linear correlation coefficient to obtain the coefficient of determination works only for the least-squares linear regression model line y ̂ = b o + b 1 x

58 Summary The coefficient of determination r 2 measures the proportion of the variation in the outcome variable that is explained by the least-squares regression line. The larger the value of r 2 , the closer the predictions made by the least-squares regression line are to the actual values, on average. To compute the coefficient of determination, first compute the correlation coefficient, then square it to obtain r 2 . (Only works for least-square linear regression model.) Regression Examples using the calculator 1) A random sample of 11 statistics students produced the following data, where x is the third exam score out of 80, and y is the final exam score out of 200. Can you predict the final exam score of a random student if you know the third exam score? Third exam score 65 67 71 71 66 75 67 70 71 69 69 Final exam score 175 184 187 185 170 198 183 182 185 180 178 a) What is the explanatory variable? ________________ What is the response variable? ____________ b) What is the least squares regression equation? _______________________________ c) Interpret the meaning of the slope. d) What is the correlation coefficient, r? ___________ Interpret this result. . e) What is the coefficient of determination, r 2 ? ___________Interpret this result f) Suppose you received a score of 72 on the third exam, predict the score on the final exam.

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59 2) SCUBA divers have maximum dive times they cannot exceed when going to different depths. The table below shows different depths with the maximum dive times in minutes. Depth in feet 50 60 70 80 90 Maximum dive time 80 55 45 35 25 a) What is the explanatory variable? ________________ What is the response variable? _____________ b) What is the least squares regression equation? _____________________________ c) Interpret the meaning of the slope. d) What is the correlation coefficient, r? ___________ Interpret this result. . e) What is the coefficient of determination, r 2 ?__________ Interpret this result f) Predict the maximum dive time for a) 85 feet and b) 110 feet. 3) The following table shows the life expectancy for an individual born in the United States in certain years. Year of Birth 1930 1940 1950 1965 1973 1982 1987 1992 2010 Life Expectancy 59.7 62.9 70.2 69.7 71.4 74.5 75 75.7 78.7 a) What is the explanatory variable? _______________ What is the response variable? _____________ b) What is the least squares regression equation? _____________________________ c) What is the correlation coefficient, r? ___________ Interpret this result. . d) What is the coefficient of determination, r 2 ?___________ Interpret this result. e) Predict the life expectancy of an individual born in the United States in the year 2000.

60 4) The following table gives the gold medal ti mes for every other Summer Olympics for the women’s 100-meter freestyle (swimming). Year 1912 1924 1932 1952 1960 1968 1976 1984 1992 2000 2008 2016 Time (seconds) 82.2 72.4 66.8 66.8 61.2 60.0 55.65 55.92 54.64 53.8 53.1 52.7 a) What is the explanatory variable? _______________ What is the response variable? _____________ b) What is the least squares regression equation? _____________________________ c) What is the correlation coefficient, r? ___________Interpret this result. . d) What is the coefficient of determination, r 2 ? _________Interpret this result.

61 Minitab Regression Examples 1) The Kelley Blue Book provides information on wholesale and retail prices of cars. The Minitab printout below presents the years old the car is and the price in dollars. Determine the following: a) What is the explanatory variable?______________ What is the response variable? _____________ b) What is the least squares regression equation?_________________________ c) What is the coefficient of determination, r 2 ?_______________ Interpret this result. d) What is the correlation coefficient, r? ____________ Interpret this result. e) Are there any outliers or influential observations? If so, which observations are outliers? _________ which are influential?______________ f ) Suppose the age of the cars ranged from 1 to 11 years. What would be the predicted price if the car that is 3.75 years old?_______________________ What would be the predicted price if the car if it was 15 years old?___________________ Regression Analysis: Price versus Age Regression Equation Price = 20550 - 1580 Age Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 20550 639 32.17 0.000 Age -1580 135 -11.71 0.000 1.00 Model Summary S R-sq R-sq(adj) R-sq(pred) 2942.12 61.46% 61.01% 59.61% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 1187289816 1187289816 137.16 0.000 Age 1 1187289816 1187289816 137.16 0.000 Error 86 744423496 8656087 Lack-of-Fit 9 157482746 17498083 2.30 0.024 Pure Error 77 586940750 7622607 Total 87 1931713312 Fits and Diagnostics for Unusual Observations Obs Price Fit Resid Std Resid 10 7987 3170 4817 1.74 X 27 4995 4750 245 0.09 X 40 24675 14230 10445 3.57 R 42 4321 11070 -6749 -2.32 R 81 22995 9490 13505 4.66 R R Large residual X Unusual X

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62 2) Is the average a teacher gets pay and the amount spent per student in each of the 50 states and the District of Columbia have a linear relationship? The Minitab printout below presents the average teacher salary and the average amount spent per student. Determine the following: a) What is the explanatory variable? ______________ What is the response variable?____________ b) What is the least squares regression equation?__________________________ c) What is the coefficient of determination, r 2 ? _____________Interpret this result. d) What is the correlation coefficient, r?_____________ Interpret this result. e) Are there any outliers or influential observations? If so, which observations are outliers?_________ which are influential?________________ f) Suppose the amount spent per student was between $8,097 and $22,366. What would the predict school teacher annual salary to be: 1) if the amount spent per student was $24,000? __________________________ 2) if the amount spent per student was $12,500?_____________________________ Regression Analysis: Average Pay of Teacher versus spending per student Regression Equation Average Pay of Teacher = 22074 + 1.318 spending per student Coefficients Term Coef SE Coef T- Value P- Value VIF Constant 22074 1635 13.50 0.000 spending per student 1.318 0.123 10.73 0.000 1.00 Model Summary S R-sq R-sq(adj) R-sq(pred) 2852.39 70.14% 69.53% 65.17% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 936569507 936569507 115.11 0.000 spending per student 1 936569507 936569507 115.11 0.000 Error 49 398670200 8136127 Total 50 1335239707 Fits and Diagnostics for Unusual Observations Obs Average Pay of Teacher Fit Resid Std Resid 9 55209 47334 7875 2.90 R 12 46790 40199 6591 2.34 R 31 51443 50916 527 0.20 X 33 45589 52880 -7291 -2.90 R X R Large residual X Unusual X

63 Section I: Homework 1) A researcher wishes to determine if there is any correlation between the scores on a reading test with those on a writing test, in other words does writing ability depend on reading ability? He gives both tests to seven people selected at random and the scores are recorded below. Reading Test 6 6 5 3 2 2 0 Writing Test 10 8 8 6 5 3 1 a) What is the explanatory variable? ________________ What is the response variable? _____________ b) What is the least squares regression equation? _______________________________ c) What is the correlation coefficient, r?_____________ Interpret this result. d) What is the coefficient of determination, r 2 ? ___________Interpret this result e) Suppose you received a score of 4 on the reading test, predict the score on the writing test._________ 2) A teacher wishes to determine if there is a correlation between her students’ computational ability and mathematical creativity, two tests are given to 5 students selected at random and the scores are recorded below. Does mathematical creativity depend on computational ability? Computational ability 48 46 46 44 43 Mathematical creativity 42 43 44 46 47 a) What is the explanatory variable? ________________ What is the response variable? ____________ b) What is the least squares regression equation? _______________________________ c) What is the correlation coefficient, r? ___________ Interpret this result. . d) What is the coefficient of determination, r 2 ?___________ Interpret this result e) Suppose you received a score of 45 on the computational ability test, predict the score on the mathematical creativity test. ___________

64 3) A teacher wants to see if there is a relationship between math aptitude and language aptitude. She takes a random sample of 6 students and looks at their aptitude scores. Does language aptitude depend on math aptitude? Math Aptitude 525 515 510 495 430 400 Language Aptitude 550 535 535 520 455 420 a) What is the explanatory variable? ________________ What is the response variable? _____________ b) What is the least squares regression equation? _______________________________ c) What is the correlation coefficient, r? ___________ Interpret this result. . d) What is the coefficient of determination, r 2 ?___________ Interpret this result e) Suppose a student had a math aptitude score of 500, predict the student’s language aptitude sc ore. __________ 4) A random sample of men were stopped in a shopping center and asked their shoe size and the number of ties that they owned. Would you expect there to be any correlation between the 2 variables? Does number of ties depend on shoe size? Shoe size 7.5 9 9 11 8.5 8 13 10 10 10 Number of Ties Owned 10 17 17 4 10 1 6 9 11 10 a) What is the explanatory variable? ________________ What is the response variable? _____________ b) What is the least squares regression equation? _______________________________ c) What is the correlation coefficient, r? ___________ Interpret this result. . d) What is the coefficient of determination, r 2 ?___________ Interpret this result e) If a man has a shoe size of 8 what would be the predicted number of ties he owns? ________ f) Is there a correlation between the two variables?__________ Explain.

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65 5) Researchers want to determine if there is a relationship between systolic blood pressure and weight. A random sample of 10 patients’ systolic blood pressure and weight were analyzed and the results are in the Minitab printout below. Regression Analysis: Systolic versus weight Regression Equation Systolic = 1.1 + 0.7638 weight Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 1.1 12.4 0.09 0.928 weight 0.7638 0.0632 12.09 0.000 1.00 Model Summary _____S R-sq R-sq(adj) R-sq(pred) 3.45944 94.20% 93.56% 93.41% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 1749.20 1749.20 146.16 0.000 Error 9 107.71 11.97 Total 10 1856.91 Fits and Diagnostics for Unusual Observations Obs Systolic Fit Resid Std Resid 7 137.00 144.74 -7.74 -2.37 R a) What is the explanatory variable?______________ What is the response variable?_____________ b) What is the least squares regression equation? ________________________________________ c) What is the coefficient of determination, r 2 ? __________________Interpret this result. d) What is the correlation coefficient, r? ________________ Interpret this result. e) Are there any outliers or influential observations? _______________ If so, which observations are outliers?______________ which are influential?_______________ f ) Suppose the weights ranged from 167 to 217 pounds. What would be the predicted systolic blood pressure if the weight is 187 pounds? ______________ What would be the predicted systolic blood pressure if the weight is 250 pounds?_______________

66 6) Researchers were testing the ground water in Northwest Texas to test the pH and the amount of bicarbonate in well water. A random sample of 33 wells in Northwest Texas were tested. Is there a relationship between the pH and the amount of bicarbonate (parts per million) in the well water? ( Reference: Union Carbide Technical Report K/UR-1) Regression Analysis: pH versus Bicarbonate Regression Equation pH = 8.098 - 0.00305 Bicarbonate Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 8.098 0.229 35.40 0.000 Bicarbonate -0.00305 0.00149 -2.04 0.049 1.00 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.479047 11.53% 8.76% 1.40% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 0.9568 0.9568 4.17 0.049 Error 32 7.3435 0.2295 Total 33 8.3003 Fits and Diagnostics for Unusual Observations Obs pH Fit Resid Std Resid 22 8.800 7.649 1.151 2.44 R 32 6.700 7.740 -1.040 -2.21 R X a) What is the explanatory variable?______________ What is the response variable?_____________ b) What is the least squares regression equation? ________________________________________ c) What is the coefficient of determination, r 2 ? __________________ Interpret this result. d) What is the correlation coefficient, r? ________________ Interpret this result. e) Are there any outliers or influential observations? _______________ If so, which observations are outliers?______________ which are influential?_______________ f ) Suppose the amount of bicarbonate ranged from 35 to 215 parts per million. What would be the predicted pH if the amount of bicarbonate is 28 parts per million? ___________ What would be the predicted pH if the amount of bicarbonate is 126 parts per million? __________

67 7) A group of movie buffs wanted to determine if there was a relationship between box office receipts during the first year and the total production cost of a movie. Regression Analysis: box office receipts versus Total production cost Regression Equation box office receipts = 15.5 + 7.98 total production cost Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 15.5 11.6 1.34 0.218 Total production cost 7.98 1.22 6.52 0.000 1.00 Model Summary S R-sq R-sq(adj) R-sq(pred) 14.2578 84.17% 82.19% 75.48% Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Regression 1 8647 8647.4 42.54 0.000 Error 8 1626 203.3 Total 9 10274 Fits and Diagnostics for Unusual Observations Obs Box office receipts Fit Resid Std Resid 4 130.60 100.88 29.72 2.23 R 8 79.30 70.76 34.78 1.45 X a) What is the explanatory variable?______________ What is the response variable?_____________ b) What is the least squares regression equation? ________________________________________ c) What is the coefficient of determination, r 2 ? __________________ Interpret this result. d) What is the correlation coefficient, r? ________________ Interpret this result. e) Are there any outliers or influential observations? _______________ If so, which observations are outliers?______________ which are influential?_______________ f ) Suppose the amount of total production costs were between 3.1 and 15.1 million dollars. What would be the predicted box office receipts if the total production costs were 4.5 million dollars? ________ What would be the predicted box office receipts if the total production costs were 25.4 million dollars? _______

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68 Section J Basic Probability Concepts Before we can begin to discuss inferential statistics, we need to discuss probability. Recall, inferential statistics deals with analyzing a sample from the population to draw conclusions about the population, therefore since the data came from a sample we can never be 100% certain the conclusion is correct. Therefore, probability is an integral part of inferential statistics and needs to be studied before starting the discussion on inferential statistics. Theoretical probability is the theory behind probability. To find the probability of an event using theoretical probability, actually conducting an experiment is not necessary. You need to use your knowledge about the situation, some reasoning and/or known formula to calculate the probability of an event happening. It can be written as the ratio of the number of favorable events divided by the number of possible events. Law of Large Numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability. A sample space contains all possible outcomes of a probability experiment. EX: An event is an outcome or a collection of outcomes from a sample space. A probability model for a probability experiment consists of a sample space, along with a probability for each event. Note: If A denotes an event then the probability of the event A is denoted P(A). Probability models with equally likely outcomes If a sample space has n equally likely outcomes, and an event A has k outcomes, then P(A) = Number of outcomes in A Number of outcomes in the sample space = k n The probability of an event is always between 0 and 1, inclusive.

69 Important probability characteristics: 1) For any event A, 0 ≤ P(A) ≤ 1 2) If A cannot occur, then P(A) = 0. 3) If A is certain to occur, then P(A) = 1 An unusual event is one whose probability is small. Basically, any probability less than 0.05 would be considered unusual. Example : Using a Tree Diagram for Finding a Sample Space Example: Find the sample space for having three children

70 Sampling from a Population is a Probability Experiment Sampling an individual from a population is a probability experiment. The population is the sample space and the members of the population are equally likely outcomes. Examples: 1) The following table shows the results of a survey of college freshman, asking how much they pay out-of-pocket, as a student, per year for college. a) P(student pays between $55,000 and $65,000) = b) P(student pays $65,000 or over) = c) P(student pays under $25,000) = d) P(student pays under $5,000) = e) P(student pays $5,000 or more) = 2) In a survey of 400 likely voters in a certain city, 215 said that they planned to vote to reelect the incumbent governor. a) What is the probability that a surveyed voter plans to vote to reelect the incumbent governor? b) What is the probability that a surveyed voter plans to vote for a new governor? 3) During a recent softball season, a softball pitcher threw 125 fast balls, 242 rise balls, 228 drops and 236 curve balls. a) What is the probability the softball pitcher threw a rise ball? b) What is the probability the softball pitcher threw a fast ball? c) What is the probability the softball pitcher threw a curve ball or a drop?

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71 Empirical Probability or experimental probability of an event – consists of repeating an actual experiment a large number of times and finding the ratio (decimal or fraction) of the number of outcomes in which a specified event occurs to the total number of trial. Examples 1) A fair 6-sided die is rolled 800 times. On 140 of those rolls the die comes up 3, on 128 of those rolls the die comes up 4 and on 135 of those rolls the die comes up 1. a) Find the empirical probability the die will come up 3. b) Find the empirical probability the die will come up 4. c) Find the empirical probability the die will come up 1. d) Find the empirical probability the die will come up any number except 1. 2) Two dice are rolled 450 times, the sum of 7 comes up 80 times, the sum of 6 comes up 65 times and the sum of 12 comes up 12 times. a) Find the empirical probability the sum of the dice will be 7. b) Find the empirical probability the sum of the dice will be 6. c) Find the empirical probability the sum of the dice will be 12.

72 Section J: Homework (Write probabilities as decimals rounded to two decimal places.) 1) Assume we have an ordinary deck of 52 playing cards. Deck of Cards: Black: Clubs Black: Spades Red: Hearts Red: Diamonds Find the following probabilities: a) P(spade) = _____________ b) P(jack) = ___________ c) P(diamond or club) = __________ d) P(queen or king) = __________ e) P(King of diamonds) = ________ f) P(picture card) = _________ 2) Assume we have a slot machine with three wheels. Each wheel has a picture of one apple and one cherry. So for example on a particular play we might see ACA on the wheels where A stands for apple and C for cherry. Answer the following questions: a) List the 8 distinct possibilities for the results of playing the slot machine. (Note: ACA is one of the 8) b) P(exactly one apple) = ___________ c) P(at least two apples) = _____________ d) P(at most one cherry) = ___________ e) P(the same fruit on all three wheels) = ___________ f) P(not getting three apples) = _____________ g) If we play the machine 72 times about how many times should we get the same fruit on all three wheels?__________

73 3) The following is the results of a survey of students. The students were asked what division their major was in. A student is selected at random, find the following probabilities: a) P(student is in Health Professions) = ___________ b) P(student is in Business) = __________ c) P(student is in mathematics or computer science) = _______ d) P(student is in education) = ____________ e) P(students is in mathematics or science) = ___________ 4) A carnival spinning wheel has the numbers 1 to 20 on it, all equally marked off. When it is spun, it will stop, at random, on one of the numbers. a) What is the probability that it will stop on the number 14?____________ b) What is the probability that it will stop on an even number?___________ c) What is the probability that it will stop on a number equal to 15 or higher?_______________ 5) The following data was obtained from the U.S. Census Bureau, from the 2010 Census. Using these figures, if a person is picked at random for the U.S. population in the year 2017, find the following probabilities: a) P(the person is under 18 years old) = __________ b) P(the person is at least 65 years old) =___________ c) P(the person is 18 to 44 years old) = ___________ 6) In a survey of 450 students, 275 said they would rather run the mile instead of swimming 500 yards for gym class. a) What is the probability that a surveyed student would rather run in gym class?___________ b) What is the probability that a surveyed student would rather swim in gym class?__________

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74 7) During a recent basketball season, a player made 124 2-point shots, 37 3-point shots and 75 foul shots. a) What is the probability the player made a foul shot?________________ b) What is the probability the player made a 2-point shot?____________ c) What is the probability the player made a 2-point shot or a 3-point shot?___________ 8) A fair coin is tossed 1000 times. On 550 of those flips the coin comes up heads. a) Find the empirical probability the coin will come up heads. b) Find the empirical probability the coin will come up tails.

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75 Section K The Addition Rule and the Rule of Complements In the previous section, the probabilities found were just for one event, now we will look at how to find probabilities for two or more events together, in other words compound events. (Round all answers to two or three decimal places.) A compound event is an event that is formed by combining 2 or more events. P(A or B) = P(A occurs or B occurs or both occur) – inclusive “or” P(A and B) = P(both A and B occur) Contingency Table – a table showing the distribution of one variable in rows and another in columns. Examples: 1) The following table shows the results of a survey for the income level and an individual’s favorite form of entertainment. Income Favorite Form of Entertainment Television Movies Theatre (live) Total Under $25,000 35 20 5 60 Between $25,000 and $50,000 25 18 7 50 Over $50,000 12 14 14 40 Total 72 52 26 150 A person is selected at random from this group, calculate the following probabilities: a) Find the probability that a randomly chosen individual’s favorite form of entertainment is going to the movies. P(Movies) = b) P(Income is under $25,000) = c) P(Income is over $50,000 or favorite form of entertainment is going to the Theatre) = d) P(Income between $25,000 and $50,000 and favorite form of entertainment is going to the movies) = e) P(Income is over $25,000) =

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76 2) The following table shows the results of a survey dealing with age and gambling. A person is selected at random from this group, calculate the following probabilities: a) P(The person gambles occasionally) = b) P(The person is aged between 21 and 30 or never gambles) = c) P(The person is over 45 and gambles frequency) = d) P(The person is over 31) = e) P(The person gambles frequency or occasionally) = f) P(The person is not under 20) = In the above examples, since you are given a contingency table you do not need to use formulas to find probabilities, but you are not always given a contingency table so formulas are needed to find certain probabilities. The General Addition Rule For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) For any two events A and B, P(A and B) = P(A) + P(B) – P(A or B) Examples: 3) If P(A) = 0.35, P(B) = 0.80 and P(A and B) = 0.25. Find P(A or B). 4) If P(A) = 0.46, P(B) = 0.62 and P(A or B) = 0.73. Find P(A and B). Age Gambling Frequently Occasionally Never Total Under 20 12 18 20 50 21 – 30 10 17 23 50 31 – 45 28 15 7 50 Over 45 10 10 30 50 Total 60 60 80 200

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77 Two events are mutually exclusive if it is impossible for both events to occur. P(A and B) = 0 If A and B are mutually exclusive events , then P(A or B) = P(A) + P(B) Examples: 5) If P(A) = 0.4, P(B) = 0.3, and A and B are mutually exclusive. Find P(A or B). 6) If P(A) = 0.7, P(B) = 0.2, and P(A or B) = 0.9. Are A and B mutually exclusive? 7) If P(A) = 0.35, P(B) = 0.45 and P(A or B) = 0.7. Are A and B mutually exclusive?

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78 Complement If A is any event, the complement of A is the event that A does not occur. The complement of A is denoted A C . Note: P(A) + P(A C ) = 1, so P(A C ) = 1 – P(A) Example: 8) If P(A) = 0.25 and P(B) = 0.45. Find P(A C ) and P(B C ). More examples: 9) A survey of type of accommodation a person lives in resulted in the following table: A person is selected at random. Find the following probabilities: a) P(the person lives in a Condo) = b) P(the person lives in a House) = c) P(the person lives in an apartment or a townhouse) = Type of Accommodation Frequency House 468 Condo 279 Apartment 646 Townhouse 343 Total 1736

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79 10) Let B be the event that a car brought in for service needs new brakes and let S be the event the car needs new struts. Suppose that P(B) = 0.20, P(S) = 0.15 and P(B and S) = 0.05. a) Find the probability the car needs brakes or struts or both. b) Find the probability the car does not need new brakes. 11) Last semester at Mercer, 250 students enrolled in both MAT125 and ENG101. Of these students 38 earned an A in statistics, 50 earned an A in English and 20 earned an A in both statistics and English. a) Find the probability a randomly chosen student earned an A in MAT125 or ENG101 or both. b) Find the probability a randomly chosen student did not earn an A in MAT125. 12) In a BIO103: Anatomy and Physiology class there were 40 students. 23 were females and 17 were males. Three males and six females earned an A in the course. A student is chosen at random from the class. a) Find the probability the student is a male. b) Find the probability the student earned an A in the course. c) Find the probability the student is male and earned an A. d) Find the probability the student is male or earned an A. e) Find the probability the student did not earn an A.

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80 13) Eight cards are in a box. The cards are numbered one through eight, respectively. The cards numbered 1,2,3,4,5 are blue and the cards numbered 6,7, 8 are red. A single card is drawn from the box at random. Find the following probabilities: a) P(card is a 2) = b) P(card is a 2 or red) = c) P(card is blue and an odd number) = d) P(card is blue or odd number) = e) P(card is odd or even) =

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81 Section K: Homework 1) The following tables shows the results of a survey given to 125 statistics students. The table shows how often a student did their homework and how well they did in the course. How often homework was completed Grade Earned Often Occasionally Never Total High 12 20 3 35 Average 40 15 5 60 Low 5 5 20 30 Total 57 40 28 125 A student is chosen at random, determine the following probabilities: a) P(Did homework Often) = ___________________ b) P(Earned an average grade for the course) = _______________ c) P(Did homework occasionally and earned an average grade) = ________________ d) P(Earned a high grade and Never did homework) = _________________ e) P(Earned a high or average grade for the course) = ______________ 2) A USA Today article titled “Yum Brands builds dynasty in China” (February 7, 2005) reports on how Yum Brands, the world’s largest restaurant company, is bringing the fast -food industry to China, India, and other big countries. Yum Brands, a spin-off from PepsiCo. Store USA Abroad Total KFC 5454 7676 13126 Pizza Hut 6306 4680 10986 Taco Bell 5030 193 5223 Long John Silver’s 1200 33 1233 A&W All-American 485 209 694 Total 18471 12791 31262 A store is selected at random find the following probabilities, a) P(The store is located in the USA) = ____________ b) P(The store is a Pizza Hut) = __________ c) P(The store is a KFC and located Abroad) = _____________ d) P(The store is a Taco Bell or Long John Silver’s) = ___________ e) P(The store is located Abroad or is an A&W All-American) = ______________

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82 3) If P(A) = 0.26, P(B) = 0.43 and P(A and B) = 0.15. Find P(A or B). 4) If P(A) = 0.60, P(B) = .37 and P(A and B) = 0.28. Find P(A or B). 5) If P(A) = 0.49, P(B) = 0.22 and P(A or B) = 0.56. Find P(A and B). 6) If P(A) = 0.61, P(B) = 0.7 and P(A or B) = 0.85. Find P(A and B). 7) If P(A) = 0.47, P(B) = 0.23, and A and B are mutually exclusive. Find P(A or B). 8) If P(A) = 0.62, P(B) = 0.44, and P(A or B) = 0.91. Are A and B mutually exclusive? Why? 9) If P(A) = 0.19, P(B) = 0.38, and P(A or B) = 0.57. Are A and B mutually exclusive? Why? 10) If P(A) = 0.79 and P(B) = .34. Find P(A C ) and P(B C ).

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83 11) The local YMCA offers various levels of swimming lessons year-round. The March 2020 Monday and Wednesday lessons included instructions from Water Babies through Adults. The number in each classification is given in the table below. Swim Lesson Type Number of Participants Water Babies 24 Tadpoles 17 Fish 15 Flying Fish 16 Shark 12 Adult 5 Total 89 If one participant is selected at random, find the probability of the following: a) P(The participant is in Tadpoles) = __________ b) P(The participant is in Water Babies or Shark) = _______________ c) P(The participant is not in Water Babies) = _______________ 12) Let R be the event that a person likes to run and S be the event that a person likes to swim. Suppose that P(R) = 0.35, P(S) = 0.20 and P(R and S) = 0.07. a) Find the probability the person likes to run or swim. b) Find the probability the person does not like to swim. 13) An aquarium at a pet store contains 40 orange swordfish (22 females and 18 males) and 28 green swordtails (12 females and 16 males). You randomly net one of the fish. a) What is the probability that it is an orange swordfish?_________________ b) What is the probability that it is a male fish?_________________ c) What is the probability that it is an orange female swordfish?__________________ d) What is the probability that it is a female or a green swordtail?________________

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84 14) Ten cards are in a box. The cards are numbered one through 10, respectively. The cards numbered 1,2,3,4,5 are yellow and the cards 6,7,8,9,10 are green. A single card is drawn from the box at random. Find the following probabilities. a) P(card is a 5) = ______________ b) P(card is green) = ____________ c) P(card is 8 or yellow) = __________ d) P(card is odd or green) = _________ e) P(card is yellow and even) = _________ f) P(card is even or green) = __________ g) P(card is greater than 4) = __________ 15) The following table displays the 100 senators of the 117 th U. S. Congress on January 3, 2021, classified by political party affiliation and gender. Male Female Total Democrat 32 16 48 Republican 42 8 50 Independent 2 0 2 Total 76 24 100 A senator is selected at random from this group. Find the following probabilities: a) P(senator is a female Democrat) =___________ b) P(senator is a Republican or Male) = ___________ c) P(senator is a Democrat) =____________ d) P(senator is not a Democrat) = ____________ e) P(senator is an Independent) = _____________ f) P(senator is Republican or Independent) = ______________ g) P(senator is a male Independent) = _________________

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85 Section L Conditional Probability and Multiplication Rule Conditional Probability – a probability that is computed with the knowledge of additional information The conditional probability of an event B, given event A is denoted P(B | A) P(B | A) is the probability that event B occurs, given that event A occurs or has already occurred. The probability of B given A is given by 𝐏(? |?) = 𝐏(? 𝐚𝐧𝐝 ?) 𝐏(?) , where P(A) ≠ 0 You can also use: P(B |A) = Number of outcomes in A and B Number of outcomes in A This leads to the General Multiplication Rule: P(A and B) = P(A)P(B|A) or P(A and B) = P(B)P(A|B) Example: 1) Assume we have an ordinary deck of 52 playing cards. Deck of Cards: Black: Clubs Black: Spades Red: Hearts Red: Diamonds One card is chosen at random, find the following probabilities, a) P(King|Picture card) = b) P(4 of clubs|club) = c) P(2,3,4 or 5|not a picture card) = d) P(clubs|red card) = e) P(4 |black card) =

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86 2) Let A and B be events with P(A) = 0.35, P(B) = 0.25 and P(A and B) = 0.1. Find P(A|B) and P(B|A) . 3) Let A and B be events with P(A) = 0.4, P(B) = 0.6 and P(B|A) = 0.3. Find P(A and B). 4) The following table displays the 100 Senators of the 117 th U.S. Congress on January 3, 2021 viewed by political affiliation and gender. Male Female Total Democrat 32 16 48 Republican 42 8 50 Independent 2 0 2 Total 76 24 100 a) P(Senator is a Female) = b) P(Democrat) = c) P(Female and Democrat) = d) P(Female|Democrat) = e) P(Democrat|Female) = 5) At a local business, it was reported that 65 women and 74 men has college degrees. Of the women, 35 have a Master’s Degree and of the men 52 have a Master’s degree. A person who has a college degree is chosen at random, find the following probabilities: a) P(female) = b) P(Master’s Degree) = c) P(female and Master’s Degree) = d) P(Female|Master ′ s Degree) = e) P(Master ′ s Degree|Female) =

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87 Independence Two events are independent if the occurrence of one does not affect the probability the of other event occurring. In other words , 𝐏(? |?) = 𝐏(?) , the event A occurring does not affect the probability of event B occurring. If two events are not independent, then they are called dependent . Hence, if 𝐏(?|?) = 𝐏(?) , then A and B are independent , Multiplication Rule for Independent Events If A and B are independent events, then P(A and B) = P(A)P(B) Note: You can extend it : P(A and B and C and D and …) = P(A)P(B)P(C)P(D)… Examples: 6) Let A and B be independent events with P(A) = 0.6 and P(B) = 0.4. Find P(A and B). 7) Let A, B, and C be independent events with P(A) = 0.1, P(B) = 0.25 and P(C) = 0.3. Find P(A and B and C). 8) A fair coin is flipped 5 times. What is the probability that the sequence of tosses is a) HTHTH? b) HHHHH? 9) A fair die is rolled 3 times, what is the probability all 3 rolls are 1’s?

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88 Note: Mutually exclusive and independent are different concepts. Examples: 10) Let A and B be events with P(A) = 0.7, P(B) = 0.8 and P(A and B) = 0.65. a) Are A and B independent? b) Are A and B mutually exclusive? c) P(A or B) = 11) Let A and B be events with P(A) = 0.4, P(B) = 0.5 and P(A or B) = 0.9. a) Find P(A and B). b) Are A and B mutually exclusive? c) Are A and B independent? 12) Let A and B be events with P(A) = 0.6, P(B) = 0.4 and P(A or B) = 0.76 a) Find P(A and B). b) Are A and B mutually exclusive? c) Are A and B independent?

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89 13) Jimmy needs to read 7 books for his English class Fahrenheit 451 , The Great Gatsby , The Lord of the Flies , Romeo and Juliet , To Kill a Mockingbird , The Scarlett Letter , and Of Mice and Men . His teacher said the books will be read in random order. What is the probability Jimmy will read The Great Gatsby first and Of Mice and Men second. P( The Great Gatsby , then Of Mice and Men) = 14) The U.S. National Center for Education Statistics publishes information about school enrollment in Digest of Education Statistics . The table below provides information for enrollment in public and private schools levels. Type Level Public Private Total Elementary 33,903 4,640 38,543 High School 13,537 1,366 14,903 College 11,626 3,263 14,889 Total 59,066 9,269 68,335 a) P(Private school) = b) P( Private|High School) = c) Are events Private and High School independent? Explain your answer in terms of probabilities. d) Are events Private and High School mutually exclusive? Why or why not? 15) An ice chest contains 5 cans of coke, 3 cans of root beer, and 4 cans of sprite. Three cans are selected at random, without replacement. Find the following probabilities: a) P(All three cans are coke) = b) P(All three cans are sprite) = c) P(The first two cans are root beer and the third is coke) = d) P(None of the cans are coke) = e) P(the first is sprite, the second is coke and the third is root beer) =

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90 Section L: Homework 1) Let A and B be events with P(A) = 0.43, P(B) = 0.39 and P(A and B) = 0.2. Find P(A|B) and P(?|?) . 2) Let A and B be events with P(A) = 0.54, P(B) = 0.32 and P(A|B) = 0.47. Find P(A and B). 3) Let A and B be events with P(A) = 0.74 and P(B) = 0.45. Assume A and B are independent. Find P(A and B). 4) A fair coin is flipped four times. What is the probability that the sequence of flips is HTTH? 5) Let A and B be events with P(A) = 0.35, P(B) = 0.5 and P(A and B) = 0.1. a) Are A and B independent? Why? b) Find P(A or B). c) Are A and B mutually exclusive? Why?

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91 6) Let A and B be events with P(A) = 0.4, P(B) = 0.5 and P(A or B) = 0.6. a) Find P(A and B). b) Are A and B mutually exclusive? Why? c) Are A and B independent? Why? 7) The following table displays the 100 Senators of the 117 th U.S. Congress on January 3, 2021 viewed by political affiliation and gender. Male Female Total Democrat 32 16 48 Republican 42 8 50 Independent 2 0 2 Total 76 24 100 a) P(Senator is Male) = b) P(Republican) = c) P(Male and Republican) = d) P(Male|Republican) = e) P(Republican|Male) =

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92 8) According to the U.S. Census, there were 3,200,000 women employed as nurses and 330,000 men employed as nurses in 2011. Of the women, 2,528,000 were registered nurses and of the men, 31,680 were registered nurses. A nurse is chosen at random. a) What is the probability the nurse is a man? b) What is the probability the nurse is a registered nurse? c) What is the probability the nurse is a man and a registered nurse? d) P(man | registered nurse) = e) P(registered nurse | man) = 9) In a box there are 10 cards. Five are red, three are green, and two are yellow. Two cards are drawn at random from the box without replacement . Find the probability that: a) Both cards are green. b) The first card is green and the second is red. c) The first card is yellow and the second is green. d) You get a green and a yellow in any order.

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93 10) Suppose A and B are events such that P(A) = 0.57, P(B) = 0.48 and P(A or B) = 0.76. Calculate the following: a) P(A and B) = ______ b) P(A  B) = _______ c) P(B  A) = ______ d) P(A C ) =________ e) P(B C ) =_______ f) Are A and B mutually exclusive? Why or why not? g) Are A and B independent? Why or why not?

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94 Section M Discrete Probability Distribution Now that we know something about probability, we can extend the concepts that were discussed earlier in this course, specifically, relative-frequency distribution, mean and standard deviation, which we applied to variables of a finite size to other types of variables. This leads to a discussion on the basics of random variables and their probability distribution. A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted using capital letters such as X. A discrete random variable has either a finite or countable number of value; possible values can be listed. A continuous random variable has infinitely many values; possible values cannot be listed. Example: 1) Determine whether the random variable is discrete or continuous. a) The number of cars in a parking lot. b) The time you wait in line at a check out. c) The height of a building. d) The number of students in a classroom. e) The number of times you flip a coin. f) The weight of a passenger’s suitcase. The probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph, or mathematical formula. Rules for a Discrete Probability Distribution Let P(x) denote the probability that the random variable has the value x. Then 1)   1 ) ( x P and 2) 1 ) ( 0   x P This means the sum of all the probabilities in a discrete probability distribution must add up to 1 and each individual probability can never be negative or greater than 1.

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95 Examples: 2) Determine whether the table represents a discrete probability distribution. If not, explain why. a) b) c) d) x P(x) x P(x) x P(x) x P(x) 10 0.3 -2 0.15 6.2 -0.2 15 0.3 20 0.2 -1 0.35 6.3 0.4 20 0.4 30 0.2 1 0.25 6.4 0.6 25 0.5 40 0.3 2 0.25 6.5 0.2 30 0.6 3) Fill in the missing value so that the following table represents a probability distribution. A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable. It’s the same as a relative frequency histogram. The mean or expected value of a discrete random variable mean of the random variable X = 𝝁 𝑿 = ∑(𝒙 ∙ 𝑷(𝒙)) , where x is the value of the random variable and P( x ) is the probability of observing the value x . Note: the mean of a discrete random variable is thought of as the average outcome if the experiment is repeated many, many times. In other words, if a probability experiment that produces a value of a random variable is repeated over and over again, the average of the values produced will approach the mean of the random variable. Law of Large Numbers If we sample from a population, then as the sample grows larger, the sample mean will approach the population mean. Variance and standard deviation of a discrete random variable variance of the random variable X = 𝛔 𝐗 ? = ∑(? ? ∙ 𝐏(?)) − 𝛍 𝐗 ? standard deviation of the variable X = 𝛔 𝐗 = √𝛔 𝐗 ? = √∑(? ? ∙ 𝐏(?)) − 𝛍 𝐗 ?

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96 Examples: 4) Compute the mean and standard deviation of the random variable with the given discrete probability distribution. a) b) 5) The number of points scored in a domino tournament on a typical scoring play has the following probability distribution. x 5 10 15 20 25 P( x ) 0.09 0.11 0.30 0.29 0.21 a) What is the probability of scoring 10 or less? b) What is the probability of scoring 20 or more? c) What is the probability of scoring 15? d) What is the mean? e) What is the standard deviation?

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97 6) The following table defines the discrete distribution for the number of cars per household in California. a) What is the probability a California household owns 1 car? b) What is the probability a California household owns more than 2 cars? c) What is the probability a California household owns less than 3 cars? d) What is the mean? e) What is the standard deviation? 7) An insurance company sells a one-year term life insurance policy to a 80-year-old woman. The woman pays a premium of $5000. If she dies within one year, the company will pay $50,000 to her beneficiary. According to U.S. Centers for Disease Control and Prevention, the probability that a 80-year-old woman will be alive one year later is 0.9516. Let X be the profit made by the insurance company. a) Construct a probability distribution. b) Find the expected value of the profit. x P(x) Number of Cars 0 1 2 3 4 or more P(x) 0.03 0.13 0.70 0.10 0.04

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98 8) An investor is considering investing in a start-up company. She estimates that she has a probability of 0.20 of a $15,000 loss, probability of 0.35 of a $25,000 profit, probability of 0.15 of a $50,000 profit, and probability 0.30 of breaking even (a profit of $0). What is the expected value of the profit? Would you advise the investor to make the investment? Explain. x P(x) 9) You play a game with an ordinary deck of 52 cards where one card is drawn at random. If the card drawn is the ace of diamonds you win $55. If the card is any diamond other than the ace you win $10. If the card is black, you win $5. However, if you pick a heart, you lose $30. a) Construct a probability distribution and find the expected value of this game for you. x P(x) b) Is it to your advantage to play? Explain.

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99 Section M: Homework 1) Determine whether the random variable is discrete or continuous. a) The number of people attending a meeting.____________________ b) The height of a 2 year old children.______________________ c) The amount of water used to fill a swimming pool._______________ d) The number of files on a computer._____________________ e) The weight capacity of an elevator._______________________ f) The number of people on a bus.__________________________ 2) Determine whether the table represents a discrete probability distribution. If not, explain why. a) b) c) d) ____________ _____________ _____________ ______________ 3) Fill in the missing value so that the following table represents a probability distribution.

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100 4) Compute the mean and standard deviation of the random variable with the given discrete probability distribution. a) b) 5) Suppose the probability distribution for the number of years to earn a Bachelor of Science (B.S.) degree is as follows: x 3 4 5 6 7 P(x) 0.05 0.40 0.30 0.15 0.10 b) What is the probability of earning a B.S. degree in less than 5 years? _____________ b) What is the probability of earning a B.S. degree in more than 4 years?____________ c) What is the probability of earning a B.S. degree in 4 years?_______________ d) What is the mean? e) What is the standard deviation? x P(x) 1 0.2 2 0.3 3 0.3 4 0.1 5 0.1 x P(x) -10 0.14 -5 0.16 0 0.32 5 0.26 10 0.12

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101 6) Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let x = the number of children the married couple have. x 0 1 2 3 4 5 or more P(x) 0.10 0.20 0.30 0.20 0.10 0.10 a) What is the probability of a married couple having 3 or more children? _____________ b) What is the probability of a married couple having no children?____________ c) What is the probability of a married couple having less than 3 children?_______________ d) What is the mean? e) What is the standard deviation? 7) You play a game with an ordinary deck of 52 cards where one card is drawn at random. If the card drawn is the ace of hearts you win $75. If the card is any heart other than the ace you win $8. If the card is black, you win $4. However, if you pick a diamond, you lose $45. a) Construct a probability distribution and find the expected value of this game for you. x P(x) b) Is it to your advantage to play? Explain.

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102 8) An investor is considering investing in a start-up company. She estimates that she has a probability of 0.25 of a $20,000 loss, probability of 0.20 of a $35,000 profit, probability of 0.15 of a $60,000 profit, and probability 0.40 of breaking even (a profit of $0). What is the expected value of the profit? Would you advise the investor to make the investment? Why or why not? x P(x) 9) Suppose you play a game with a biased coin. The probability the coin lands on head is 0.65 and the probability 0.35. If you flip a head you win $14, if you flip a tail you lose $20. If this game is played many times is it to your advantage to play? Why? x P(x)

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103 Section N Binomial Probability Distribution One of the most important discrete random variables is the binomial random variable. Since this is a specific type of discrete random variable, we will discuss what makes a random variable binomial, learn how to use the binomial formula to calculate probabilities, learn how to use the binomial probability table to find probabilities, as well as learn how to calculate the mean and standard deviation for a binomial random variable. What makes a random variable binomial? If you are conducting an experiment in which the random variable X represents the number of successes in a sequence of trials and the following 4 conditions hold: 1. A fixed number of trials are conducted. 2. There are two possible outcomes for each trial. One is labeled “success” and the other is labeled “failure.” 3. The probability of success is the same on each trial. 4. The trials are independent. This means that the outcome of one trial does not affect the outcomes of the other trials. then X is a binomial random variable . Notation: n = number of independent trials of the experiment p = probability of success for each trial, hence 1 – p = the probability of failure X denotes the number of successes in n independent trials of the experiment. So 0 ≤ X ≤ n . Formula for Binomial Probabilities For a binomial random variable X that represents the number of successes in n trials with success probability p, the probability of obtaining x successes is 𝐏(𝐗 = ?) = ? ? (𝐩 ? )(? − 𝐩) (𝐧−?) 𝐧 where x = 0,1,2,3,…,n Note: C x = 𝑛! ?!(𝑛−?)! n , this function can be found on most calculators so you really don’t need to know this formula If X is a binomial random variable, then it has a binomial distribution with parameters n and p.

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104 Examples: 1) Determine the indicated probability for a binomial experiment with the given number of trials n and the given probability of success, p . Round answers to four decimal places . a) n = 6, p = 0.45, Find P(4). b) n = 24, p = 0.6, Find P(20). c) n = 50, p = 0.76, Find P(35). d) n = 37, p = 0.34, Find P(19). e) n = 100, p = 0.82, Find P(72). Since the binomial distribution is an important distribution in statistics, tables have been created for many of the different values of n and p and therefore very often you do not need to use the formula to calculate binomial probabilities. Table A can be used to find many binomial probabilities. This table is found the section N folder. Examples: Use Table A to find probabilities for the following problems. Round probabilities to three decimal places . 2) A student takes a multiple-choice test that has 15 questions. Each question has four choices. The student forgot about the test and decides to guesses randomly on the questions. a) n = _______ p = ________ b) Find P(x = 4). P(4) = ___________ c) Find P(x is more than 4). P(x > 4) = ________________

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105 3) A bent coin has a probability of landing on heads equal to 0.40. This coin is tossed 5 times. a) n = _______ p = __________ b) What is the probability of getting at least 3 heads? P(X ≥ 3) = ______________ c) What is the probability of getting at most 3 heads? P(X ≤ 3) = ______________ 4) Assistant Professor Ratso, a leading experimental psychologist, is in the habit of sending mice through mazes. She predicts that a mouse reaching the end of a T-shaped maze is more likely to turn left than right. She believes that the proportion of mice which turn left is 0.70. If this is true and she sends 9 mice down the maze, what is the probability that a) n = _________ p = _________ b) exactly 3 will turn left. c) Less than 5 will turn left. d) All the mice turn left. 5) Allison Wonderland conducted a survey and found that 25% of adults read between one and five books last year. In a random sample of 14 adults, what is the probability that a) n = ___________ p = __________ b) Exactly 5 adults read between one and five books last year? c) Fewer than 3 adults read between one and five books last year? d) At least 3 adults read between one and five books last year? e) Between 5 and 7 adults, inclusive, read between one and few books last year?

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106 Mean (expected value) and standard deviation of a binomial random variable Let X be a binomial random variable with n trials and success probability p. Then the mean of X = 𝛍 ? = 𝐧𝐩 and standard deviation of X = 𝛔 𝐗 = √𝐧𝐩(? − 𝐩) Note: For a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes bell-shaped. As a rule of thumb, if np(1 – p) ≥ 10, the probability distribution will be approximately bell-shaped. Examples: 6) Allison Wonderland conducted a survey and found that 25% of adults read between one and five books last year. In a simple random sample of 200 households, determine the mean and standard deviation for the number of adults who read between one and five books last year. n = __________ p = __________ 7) According to AirHelp, only 64% of flights departed on time from Newark Airport in 2019. Suppose 100 flights are randomly selected from Newark Airport, determine the mean and standard deviation for the number of on time flights. n = __________ p = __________ 8) According to the U.S. Census Bureau, the percentage of the population 25 years and older earning at least a bachelor’s degree has increased over the last 15 years. It was determined that 35% of people 25 years and older earned at least a bachelors’ degree. In a random sample of 500 people 25 years and older, compute the mean and standard deviation for the number of people who have earned at least a bachelor’s degree . n = __________ p = __________ 9) According to the National Survey on Drug Use and Health, 80% of all adult smokers begin smoking by age 18. Compute the mean and standard deviation of the number of smokers who began smoking by 18 years of age in 200 trials of a probability experiment. n = __________ p = __________

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107 Section N: Homework 1) Determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p . Use the binomial probability distribution formula. a) n = 7, p = 0.74, Find P(6). b) n = 27, p = 0.67, Find P(25). c) n = 65, p = 0.43, Find P(35). d) n = 32, p = 0.34, Find P(19). e) n = 200, p = 0.82, Find P(165).

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108 Use Table A to answer the following: 2) A student takes a multiple-choice test that has 20 questions. Each question has five choices. The student forgot about the test and decides to guess randomly on the questions. a) State why this is a binomial distribution. State n and p. b) Find P(4) = _____________ c) P(11) = ________________ d) Find P(More than 4 questions are answered correctly) = __________ e) Find P(Less than 5 questions are answered correctly) = __________ f) What is the mean number of questions answered correctly? ______________________ g) What is the standard deviation for the number of questions answered correctly? ____________ 3) A bent coin has a probability of landing on heads equal to 0.6. This coin is tossed 8 times. a) State why this is a binomial distribution. State n and p. b) What is the probability of getting exactly 8 heads?___________________ c) What is the probability of getting at least 5 heads?____________________________________ d) What is the probability of getting at most 3 heads?___________________________________ e) What is the mean number of heads?_______________________________________________ f) What is the standard deviation?___________________________________________________

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109 4) A nurse comments that when a patient calls the medical advice line claiming to have the flu, the chance he or she truly has the flu (and just a nasty cold) is only about 10%. Of the next 20 patients calling in claiming to have the flu, we are interested in how many actually have the flu. a) State why this is a binomial distribution. State n and p. b) What is the probability at least 6 patients calling have the flu?______________________ c) What is the probability exactly 5 patients calling have the flu?______________________ d) What is the probability at most 3 patients calling have the flu?______________________ e) What is the mean number of patients calling truly have the flu?_____________________ f) What is the standard deviation?_______________________________ 5) A survey goes out to 15 college seniors to see if they will attend graduation ceremonies. Based on past years, it is known that 75% of seniors attend graduation ceremonies. a) State why this is a binomial distribution. b) What is the probability that 12 or more students will attend graduation ceremonies?_________ c) What is the probability that less than 8 students will attend graduation ceremonies?_________ d) What is the probability exactly 12 students will attend graduation ceremonies?_____________ e) What is the mean number of students who will attend graduation ceremonies?_____________ f) What is the standard deviation?____________________

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110 6) Seventy percent of people pass the state driver’s exam on the first try. A group of 150 individuals who have taken the driver’s exam is randomly selected. Determine the mean and standard deviation for the number of individuals who pass the state driv er’s exam on the first try. n = ______ p = ______ 7) Approximately, 97% of all colleges and universities in the United States offer at least one online course. A random sample of 550 was selected. Determine the mean and standard deviation for the number of colleges and universities in the United States that offer at least one online course. n = ______ p = ______

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MAT125 AllofUnit2

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