Stat281 Noteguides

Section 1.2 ______________________ - is the entire collection of individuals or objects that you want to learn about. ______________________ - is part of the population that is selected for study. What are the population and sample in our Wheat example? In an ____________________________________________ , the person carrying out the study does not control who (or what) is in which population. It is important to obtain samples that are ________________________ of the corresponding populations. In an ____________________________________________ , the person carrying out the study does control who (or what) is in which experimental group. Is the Wheat example an observational study or an experiment and why? Section 1.3 ______________________________________________ is a number that describes the entire population. ______________________________________________ is a number that describes a sample. In our example, is the data that we collected a Population Characteristic (Parameter) or a Statistic? Explain.

Stat 281 Ch. 2.1 – 2.3 Stat 281 Survey Data. I have just copied a portion of the data here. The full set will be on D2L. Stat 281 Survey Data Spring 2015 What can we observe from the survey data? Device Time Country Region Semester Gender Reason Year School Weight Height State Age Colleg DESKTOP/LAP 82 US SD 1 M 1 1 115 65 6 19 2 DESKTOP/LAP 89 US SD 5 M 1 3 120 68 41 21 2 DESKTOP/LAP 72 US SD 1 M 1 1 130 71 41 18 6 SMARTPHONE 104 US SD 1 M 1 1 130 68 23 19 6 DESKTOP/LAP 75 US SD 2 M 1 2 135 68 41 19 1 DESKTOP/LAP 97 US SD 1 M 1 2 135 70 47 18 1 DESKTOP/LAP 65 US SD 8 M 1 4 138 68 41 22 3 DESKTOP/LAP 70 US SD 3 M 1 2 140 72 23 19 6 DESKTOP/LAP 96 US SD 3 M 1 2 148 69 41 21 7 DESKTOP/LAP 131 US SD 3 M 1 2 150 67 15 21 1 SMARTPHONE 85 US IA M 1 4 160 79 23 21 4 DESKTOP/LAP 219 US SD 1 M 3 1 160 76 23 19 1 DESKTOP/LAP 56 4 M 1 3 160 74 41 21 6 DESKTOP/LAP 55 US SD 3 M 1 2 160 74 23 20 1 DESKTOP/LAP 92 US SD 5 M 1 3 160 76 41 21 1 DESKTOP/LAP 52 US SD 5 M 3 3 163 69 5 21 1 DESKTOP/LAP 111 US SD 1 M 1 1 165 72 19 5 DESKTOP/LAP 58 US SD 1 M 1 2 165 71 23 19 1 DESKTOP/LAP 88 US SD 2 M 2 2 167 72 2 22 2 DESKTOP/LAP 72 US SD 3 M 1 2 170 68 41 20 1 DESKTOP/LAP 52 US SD 5 M 1 3 170 71 41 20 1 DESKTOP/LAP 47 US SD 15 M 2 4 171 28 2 DESKTOP/LAP 87 US SD 3 M 1 2 175 75 41 20 1 DESKTOP/LAP 98 US SD 6 M 1 3 175 72 41 21 6 DESKTOP/LAP 77 US SD 1 M 1 1 175 72 41 19 1 DESKTOP/LAP 64 US SD 3 M 1 2 175 68 23 20 1 DESKTOP/LAP 82 US SD 3 M 1 2 175 73 41 19 6 DESKTOP/LAP 72 US SD 3 M 2 2 175 72 41 20 1 DESKTOP/LAP 64 US SD 8 M 1 3 175 70 23 21 6 DESKTOP/LAP 129 US SD 4 M 1 3 175 72 23 24 7 DESKTOP/LAP 55 US SD 7 M 1 5 180 80 43 25 6 DESKTOP/LAP 58 US SD 1 M 3 2 180 72 41 19 1 DESKTOP/LAP 62 US SD 5 M 1 3 183 73 41 21 2 DESKTOP/LAP 62 US SD 1 M 1 2 185 72 15 19 1 DESKTOP/LAP 104 US SD 3 M 1 2 185 72 15 20 1 DESKTOP/LAP 194 US SD 2 M 1 2 185 72 25 24 3 DESKTOP/LAP 37 US SD M 1 5 185 6 41 36 4 DESKTOP/LAP 62 US SD 4 M 1 2 187 72 41 20 5 DESKTOP/LAP 87 US SD 3 M 1 2 190 74 23 19 6 SMARTPHONE 106 US SD 4 M 1 2 195 75 23 20 1 SMARTPHONE 54 US SD 3 M 1 2 200 77 23 20 1 DESKTOP/LAP 66 US SD 4 M 1 2 200 74 41 20 1

Section 2.3. Look at the following histograms with Weight and tell me what you see. Survey Data Weight Histogram separated by Gender. Survey Data by Weight and Gender

Let’s take a look at a scatterplot on some car data from 93 cars in 1993. Car Dat with MPG and Price What are the two numerical variables and what can we learn from them? A ______________________________________ is a simple graph of data collected over time that can help you see interesting trends or patterns.

Let’s look at some misleading graphs. What is misleading about them? Number of Shingles Sold by Year

Section 3.2 Sample mean = ࠵?̅ = What is the difference between a Population Mean and a Sample Mean? Sample variance = ࠵? ଶ = Sample standard deviation = s = Population variance = ࠵? ଶ = Population standard deviation = ࠵? = Let’s look at the Spring 2015 Survey data. Now is a good time to learn some basics of Excel Weight Height Mean = Mean = Variance = Variance = Standard Deviation = Standard Deviation = Which one is more variable?

Ch. 3.4 – 3.6 Section 3.4 Summarizing a Data Set: Boxplots What are the 5 numbers that are part of the 5 number summary? 1. 2. 3. 4. 5. Here is some car data. Car MPG and Type What can we learn?

Let’s use this smaller data set to practice making a boxplot. Let’s identify the 5 numbers and make a quick sketch by hand: 2, 8, 10, 12, 16, 19, 22, 25, 28, 32, 35, 37, 44, 51, 57, 60, 64, 66, 72, 83 Then we can use the survey height data to make a boxplot in Excel. An observation can be called an outlier if it is: Section 3.5 Measures of Relative Standing: z-scores and Percentiles The _______________ tells you how many standard deviations the data value is from the mean. The z-score is positive when the data value is ___________________ than the mean and negative when the data value is __________________ than the mean. z-score = Let x = the average age at an employee at Daktronics. Given the following information, find the z – score of an employee that is 38 years old. x-value = 38 mean = 40 standard deviation = 6 z-score =

Now given that an employee has a z-score of 1.8, what is their age? The Empirical Rule If a data distribution is mound shaped and approximately symmetric, then Approximately _____ of the observations are within 1 standard deviation of the mean. Approximately _____ of the observations are within 2 standard deviation of the mean. Approximately _____ of the observations are within 3 standard deviation of the mean. With our wheat data, what would be a 95% interval of the yield? ࠵?̅ = 58.6 ࠵? = 12.3 What about a 99.7% interval of the yield? The r th percentile is a value such that r percent of the observations in the data set fall at or _____________ that value. If your GPA was at the 75 th percentile for your class, are you near the top or the bottom of the class? Another name for the 75 th percentile is the __________________________________

Section 3.6 Avoid These Common Mistakes What’s the problem with these comparative boxplots? Survey Data with Gender and State With the following two histograms displays data with a larger standard deviation? The empirical rule only applies to distributions with a ________________ shape. Mean and standard deviation can be affected by _________________ values. If the distribution is _____________ or has _____________ the median and _______________ are often a better choice for describing center and variability.

Ch. 4.1 Chapter 4 Describing Bivariate Numerical Data Section 4.1 Correlation After we have found out what the data looks like and done some initial analysis, we may want to check and see if there is linear relationship between two variables Car Data Scatter Plot with Horsepower and Price Does there appear to be a linear correlation? For future reference r = .7882.

_____________________________________________________________ , denoted by r, measures the strength and direction of a linear relationship between two numerical variables. Properties of r 1. r is ____________________ when the linear relationship is positive and _______________________ when the linear relationship is negative. 2. The value of r is always greater than or equal to ____ and less than or equal to ______ . Strong relationships are considered when r is __________________________________. Moderate relationships are considered when r is _______________________________. Weak relationships are considered when r is __________________________________. 3. r = 1 occurs when … r = - 1 occurs when … 4. r is a measure of the extent to which x and y are ______________________ related. 5. The value of r does not depend on the unit of ________________________________ for either variable.

Try to predict r for the following: 20 30 40 10 15 20 25 30 35 cty hwy 20 30 40 2 3 4 5 6 7 displ hwy

Calculating the value of the correlation coefficient r. ࠵? ௫ = ࠵? ௬ = The correlation coefficient r = Let’s try an example. How much should a healthy Shetland pony weigh? Let x be the age of the pony (in months), and let y be the average weight of the pony (in kilograms). X 3 6 12 18 24 y 60 95 140 170 185 Compute r and round answer to 3 decimal places.

Lemons Imported and US Highway Fatality Rate In this example r = - .985. Do you believe that the increase in lemons being imported has caused a decrease in the highway fatality rate? Just because there is a strong correlation coefficient does not mean that there is a cause and effect relationship!

Ch. 4.2 Wheat data taken from SDSU about the Yield and Protein levels taken 2012 – 2014 Section 4.2 Linear Regression: Fitting a Line to Bivariate Data When there is a linear relationship between two variables, you can use information about one variable to learn about the value of the second variable. The letter y is used to denote the variable you would like to predict, and this variable is called the _____________________________ (dependent variable). The other variable, denoted by x, is the ____________________________________________ (independent or explanatory variable). The equation of a line y = a + bx, where b = ______________ , and a = ___________________ . The least squares regression line is the line that ___________________________ the sum of squared deviations. The slope of the least squares regression line is b = Calculating formula for slope that is easier for calculations is b = The y – intercept is a = The equation for the least squares regression line is ࠵? ෝ =

Go back to our example of Yield and Protein which had r = - .7182. Let’s say that we want to come up with a line of best fit for that data. Coefficients: (Intercept) YIELD 17.63036 -0.03934 b = a = ࠵? ෝ = If Yield = x = 59.7, then what would our estimated value for ࠵? ෝ equal? Making predictions outside the range of x values in the data set can produce misleading predictions if the pattern does not continue. This is sometimes referred to as the danger of _______________________________________________ . Refer back to the pony example in 4.1. Predict the weight of a 9-month old pony. Then think of how one could extrapolate with a line. Another example. Create a regression line relating height and weight with the survey data. Compute r as well. Let’s try an example. An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs). x 16 33 50 28 50 25 y 2 3 6 5 9 3

Total Jobs vs Entry-level Jobs in Denver Neighborhoods: Compute r. Then find the equation of the least-squares line = a + bx . For a neighborhood with x = 40 hundred jobs, how many are predicted to be entry level jobs? We will solve on 4.2 Excel example. Line of Best Fit for Total Jobs vs Entry-Level Jobs

Ch. 4.3 Assessing the Fit of a Line After we obtain the line of best fit, the next step is to determine how effective the line is at summarizing the relationship between x and y. We can consider three things: 1. 2. 3. The _________________________________ result from substituting each x value into the equation for the regression line. This gives ࠵? ො ଵ = ࠵? ො ଶ = ࠵? ො ௡ = The ___________________________________ are the n quantities ࠵? ଵ − ࠵? ො ଵ = ࠵? ଶ − ࠵? ො ଶ = ࠵? ௡ − ࠵? ො ௡ = Each residual is the difference between an observed y value and the corresponding predicted y value.

Examples of Residual Graphs Random pattern Non-random: U-shaped Non-random: Inverted U Go back to the ponies from 4.1 one more time. I will calculate the predicted values and the residuals and make a plot. Do we see any pattern here? An observation is an ___________________________ if it has a large residual. Outliers fall far away from the least squares line in the ___ direction. Alternatively, a point is potentially an ______________________________________________ if it has an x value far away from the rest of the data. The _____________________________________________________ , denoted by _____ , is the proportion of variability in y that can be attributed to an approximate linear relationship between x and y. The value of _____ is often converted to a percentage (by multiplying by 100). Residual sum of squares = SSResid = Total sum of squares = SSTo = The coefficient of determination is calculated as ࠵? ࠵? = * Note that we can calculate ࠵? ࠵? by squaring r

A large value of ࠵? ࠵? indicates that a large proportion of the variability in y can be explained by the approximate linear relationship between x and y. This tells you that knowing the value of x is helpful for predicting y. Let’s look at the famous “iris” data set. We have 150 measurements from 3 species of irises. Let’s look at the relationship of sepal lengths and widths. First we’ll look at this relationship for the whole dataset and then just for the species setosa. What is ࠵? ࠵? in each case? How much of the variability can be explained here? What is r and what does that tell us? Example of a test question: Lemon Imports vs US Highway Fatality Rate In this example, what is ࠵? ࠵? ? How much of the variability can be explained? What is r?

Section 4.4 Describing Linear Relationships and Making Predictions – Putting It All Together Let’s summarize the steps in a linear regression. 1. Summarize the data graphically by constructing a ___________________________ . 2. Based on the scatterplot, decide if it looks like the relationship between x and y is approximately __________________________. 3. Find the equation of the __________________________________________________ . 4. Construct a residual plot to look for any patterns or unusual features. ( You can usually skip this step for our class.) 5. Calculate ࠵? ࠵? and interpret. 6. Decide if the least squares line is useful in making predictions. 7. Use the least squares line to make predictions. Does there appear to be a linear relationship between carat and price? (Investigate this with the “Diamonds data set” under “Sample Data Sets”). Find the equation for the least squares regression line? What is r ? What is ࠵? ଶ ?

Interpret the slope, r , and ࠵? ଶ . Does it seem like this line will be useful to make predictions?