4.7 5.1 and part of 5.2

4.7 Linear Regression with Excel As with the calculations in the previous chapter, researchers rarely compute the correlation coefficient or the equation of the least-squares regression line by hand. Example. Use Microsoft Excel to determine the strength of the correlation between height and shoe size of MATH 1680 students. 1. Open the Microsoft Excel file named “MATH 1680 Chapter 4 Data” 2. Click on the Height vs Shoe Size sheet 3. In cell D1, type “r” in bold 4. In cell D2, enter the formula =CORREL(A:A, B:B) 5. Highlight cell D2, then select Home > Number > Decrease Decimal to display three decimal places Now use Microsoft Excel to find the equation of the least-squares regression line for height versus shoe size. 6. Highlight columns A and B 7. Select Insert > Charts > Insert Scatter (X, Y) or Bubble Chart > Scatter 8. Right-click on the vertical scale, then select Format Axis 9. Set the minimum bound to 50.0 and the maximum bound to 82.0 10. Right-click on one of the data points, then select Add Trendline 11. Check the box for Display Equation on chart 12. Check the box for Display R-squared value on chart Finally, verify the relationship between the correlation coefficient and the coefficient of determination. 13. In cell E1, type “r^2” in bold In cell E2, enter the formula =D2^2

Chapter 5 Probability 5.1 Probability Terminology and Notation Basic Definitions When discussing probability, an experiment is any process with uncertain results that can be repeated many times. Common probability experiments include:  Tossing a coin  Rolling a die  Drawing a card from a deck  Randomly selecting a sample from a population When an experiment is repeated multiple times, each repetition is called a trial . A possible result of the experiment is called an outcome . The sample space of the experiment is the set of all possible outcomes. Tossing a Coin: Heads, Tails Rolling a Die: 1, 2, 3, 4, 5, 6 Drawing a Card: A ♠ , 2 ♠ , 3 ♠, 4 ♠ , 5 ♠ , 6 ♠ , 7 ♠, 8 ♠ , 9 ♠ , 10 ♠ ,J ♠,Q♠ ,K ♠ , A ♡ , 2 ♡ , 3 ♡ , 4 ♡ , 5 ♡ , 6 ♡ , 7 ♡ , 8 ♡ , 9 ♡ , 10 ♡ ,J ♡ ,Q ♡ ,K ♡ , A ♢ , 2 ♢ , 3 ♢ , 4 ♢ , 5 ♢ , 6 ♢ , 7 ♢ , 8 ♢ , 9 ♢ , 10 ♢ ,J ♢ ,Q ♢ ,K ♢ , A ♣ , 2 ♣ , 3 ♣, 4 ♣ , 5 ♣ , 6 ♣ , 7 ♣, 8 ♣ , 9 ♣ , 10 ♣ ,J ♣,Q♣ ,K ♣ A collection of outcomes is called an event . Events can be described in words or by listing their outcomes. Example. Suppose a standard die is rolled. List all the possible ways of rolling an odd number. 1, 3, 5 Example. Suppose a coin is tossed three times. List all the possible ways of getting at least two heads. HHT, HTH, THH, HHH

If A and B are mutually exclusive events, knowing that A has occurred makes it less likely that B has also occurred. (In fact, it makes it impossible for B to occur.) Similarly, knowing that B has occurred makes it less likely for A to occur. Two events are dependent if knowing that one of the events has occurred makes it more or less likely that the other event has occurred. Two events are independent if neither event affects the likelihood of the other. Example. A coin is tossed twice. Event A is getting heads on the first toss. Event B is getting heads on the second toss. Are A and B independent? Yes Example. Two dice are rolled. Event A is the sum being greater than 10. Event B is rolling a six on the first die. Are these events independent? No Example. A student is selected at random. Event A occurs if the student is taller than 72 inches. Event B occurs if the student’s shoe size is greater than 10. Are these events dependent? Yes (Refer to a scatter plot of Height vs. Shoe Size from Chapter 4.) Example. A student is selected at random. Event A occurs if the student got more than 7 hours of sleep the night before. Event B occurs if the student’s shoe size is greater than 10. Are these events dependent? No (Refer to a scatter plot of Sleep vs. Shoe Size from Chapter 4)

Compound Events If A and B are two events, the compound event A AND B consists of all the outcomes that belong to both. Example. Two dice are rolled. Event A is rolling a six on the first die. Event B is the sum being greater than 9. List the outcomes in the event A AND B . 6+4, 6+5, 6+6 Example. Suppose that a student is selected at random. Event A occurs if the student is female. Event B occurs if the student has brown eyes. Is the event A AND B larger or smaller than the event A ? Than the event B ? “Raise your hand if you are female. Raise your hand if you have brown eyes. Raise your hand if you are female and have brown eyes.” Did the event get bigger or smaller? If A and B are two events, the compound event A OR B consists of all the outcomes that belong to either. Example. Two dice are rolled. Event A is rolling a six on the first die. Event B is the sum being greater than 9. List the outcomes in the event A OR B . 6+1, 6+2, 6+3, 6+4, 6+5, 6+6, 5+6, 5+5, 4+6 Example. Suppose that a student is selected at random. Event A occurs if the student has green eyes. Event B occurs if the student has hazel eyes. Is the event A OR B larger or smaller than the event A ? Than the event B ? “Raise your hand if you have green eyes. Raise your hand if you have hazel eyes. Raise your hand if you green or hazel eyes.” Did the event get bigger or smaller?

Example. A student is selected at random. Event F occurs if the student is female. Event B occurs if the student has brown eyes. How should we denote the probability that the student has brown eyes, given that she is female? P ( B ∨ F ) Example. Let C be the event that a randomly chosen person commutes more than 45 minutes to work. Let S be the event that a randomly chosen person lives in the suburbs. How should we denote the probability that the person commutes more than 45 minutes to work, given that they live in the suburbs? P ( C ∨ S ) Example. According to an article published by Axios , California accounts for 39% of all electric vehicles registered nationwide. However, electric vehicles only account for 2% of all vehicles registered in California. Express both facts using conditional probability. P ( California | EV ) = 0.39 and P ( EV | California ) = 0.02 5.2 Basic Probability Definitions and Basic Properties The probability of an event A , denoted P ( A ), represents the likelihood that the event will occur on a single trial of a probability experiment. The probability of an event can be expressed as a fraction, decimal, or percentage, but it must obey the following rules: 1. The probability of any event A must be between 0 and 1. 2. The probability of all possible outcomes must sum to 1. When there are n possible outcomes that are all equally likely, the probability of any particular outcome is 1/ n . More generally, suppose A is an event that includes m outcomes. Then the probability that A will occur on a single trial of the experiment is P ( A ) = m n

Example. A coin is tossed. Assuming that the coin is fair, what is the probability that it lands heads? P ( Heads ) = 1 2 Example. A fair die is rolled. What is the probability of rolling a five? P ( Five )= 1 6 Example. In a college math class, 21 students have a low level of math anxiety, 51 students have a medium level of math anxiety, and 42 students have a high level of math anxiety. What is the probability that a randomly selected student will have a high level of math anxiety? There are 21 + 51 + 42 = 114 students in the class. P ( High ) = 42 114 = 0.368 = 36.8% Example. A card is randomly drawn from a standard deck. What is the probability of drawing a face card? There are 4 * 13 = 52 cards in a standard deck, of which 12 are face cards. P ( Face ) = 12 52 = 0.231 = 23.1%

Complement Rule On a single trial of a probability experiment, either an event A or its complement A ′ must occur. Therefore, the probabilities of A and A ′ must sum to 1. P ( A ) + P ( A ' ) = 1 Subtracting the probability of A from both sides yields the formula P ( A ' ) = 1 − P ( A ) which we can use to calculate the probability of A ′ without counting the number of ways it can occur. Example. An elementary school teacher records the eye colors of her pupils. Four children have blue eyes, 19 have brown eyes, 3 have green eyes, and 2 have hazel eyes. Use the complement rule to find the probability that a randomly selected child does not have brown eyes. Since there are 28 pupils, and 19 of the pupils have brown eyes, P ( not Brown ) = 1 − 19 28 = 28 28 − 19 28 = 9 28 Example. A spinner contains the numbers in the range 1 to 50. What is the probability that the spinner will land on a number that is not a multiple of 5? 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 P ( not amultiple of 5 ) = 1 − 10 50 = 1 − 1 5 = 4 5

Example. According to CovidActNow , 70.5% of the people in Denton County have received at least one dose of a COVID-19 vaccine, and 62% of the population is fully vaccinated (two doses) as of 7/28/2022. What is the probability that a randomly selected person in Denton County is not fully vaccinated? What is the probability that they are unvaccinated? P(not Fully Vaccinated) = 1 – 0.620 = 0.380 = 38.0% P(Unvaccinated) = 1 – 0.705 = 0.295 = 29.5% Example. According to U.S. Centers for Disease Control and Prevention , in August 2020, the percentage of adults experiencing symptoms of depression during the past 7 days was 24.5%. In December 2020, the percentage of adults experiencing symptoms of depression during the past 7 days was 30.2%. Find the probability that a randomly selected adult did not experience symptoms of depression during the past 7 days in August 2020 and December 2020. P(no depression in August) = 1 – 0.245 = 0.755 = 75.5% P(no depression in December) = 1 – 0.302 = 0.698 = 69.8%