AP Stats Chapter 15. Inference for Regression
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1 AP Stats Chapter 15 Inference for Regression Example 1: Regression Review Child development researchers explored the relationship between the crying of infants and their later IQ scores. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. Crying IQ Crying IQ Crying IQ Crying IQ a) Find the regression equation for the following data set. Define any variables you use. b) Interpret the y-intercept and slope in the context of the problem. c) Find and interpret the correlation coefficient for this data. d) Find and interpret 2 R for this data.
2 When doing inference for regression we are checking to see If there is a relationship, then Picture: If there is not a relationship, then Picture: Conditions to be checked for the confidence interval and hypothesis test:
3 Check each condition for the data in example 1: In general, how do I write the hypotheses for the test? What is this hypothesis test called? What is the formula? What are the degrees of freedom? Suppose the researchers in example 1 wanted to know if there was a straight line relationship between IQ and crying. a) Write the hypotheses for this situation. b) Calculate the test statistic. What is your conclusion?
4 Example 2: Do customers who stay longer at buffets give larger tips? Charlotte, an AP statistics student who worked at an Asian buffet, decided to investigate this question for her second semester project. While she was doing her job as a hostess, she obtained a random sample of receipts, which included the length of time (in minutes) the party was in the restaurant and the amount of the tip (in dollars). Do these data provide convincing evidence that customers who stay longer give larger tips? (a) Here is a scatterplot of the data with the least-squares regression line added. Describe what this graph tells you about the relationship between the two variables. Tip (dollars) Time (minutes) Minitab output from a linear regression analysis on these data is shown below. Time (minutes) Tip (dollars) Predictor Coef SE Coef T P Constant Time (minutes) S = R-Sq = 13.2% R-Sq(adj) = 4.5% 3 2 Residual Time (minutes) (b) What is the equation of the least-squares regression line for predicting the amount of the tip from the length of the stay? Define any variables you use. (c) Interpret the slope and y intercept of the least-squares regression line in context.
5 (d) Carry out an appropriate test to answer Charlotte s question. Example 3: The local utility company surveys 101 randomly selected customers. For each survey participant, the company collects the following: annual electric bill (in dollars) and home size (in square feet). Output from a regression analysis appears below. Regression equation: Annual bill = 0.55 * Home size + 15 Predictor Coef SE Coef T P Constant Home size Is there a significant linear relationship between annual bill and home size? Use a 0.05 level of significance.
6 Example 4: Here are the data on the number of jet skis in use, the number of accidents and the number of fatalities for the years Is there a relationship between the number of accidents and the number of jet skis in use? Year No. in Use Accidents Fatalities , , , ,376 1, ,915 1, ,283 1, ,545 2, ,000 3, ,000 4, ,000 4, How do I construct a confidence interval for the slope of the regression line? Where do I get t*? What are the degrees of freedom?
7 Construct and interpret a 95% confidence interval for the slope of the regression line using the data from example 1. Example 5: For their second-semester project, two AP Statistics students decided to investigate the effect of sugar on the life of cut flowers. They went to the local grocery store and randomly selected 12 carnations. All the carnations seemed equally healthy when they were selected. When the students got home, they prepared 12 identical vases with exactly the same amount of water in each vase. They put one tablespoon of sugar in 3 vases, two tablespoons of sugar in 3 vases, and three tablespoons of sugar in 3 vases. In the remaining 3 vases, they put no sugar. After the vases were prepared and placed in the same location, the students randomly assigned one flower to each vase and observed how many hours each flower continued to look fresh. Here are the data and computer output. Predictor Coef SE Coef T P Constant Sugar (tbs) S = R-Sq = 86.0% R-Sq(adj) = 84.5% Construct and interpret a 99% confidence interval for the slope of the true regression line. Sugar (tbs.) Freshness (hours)
8 Example 6: Does how long young children spend at the lunch table determine how much they eat? Here are data on 20 toddlers observed over several months at a preschool. Make a scatterplot of the data and find the equation of the least squares regression line for predicting calories consumed from time at the table. Then construct a 95% confidence interval for the slope of the true regression line. Time: Calories: Example 7: The local utility company surveys 101 randomly selected customers. For each survey participant, the company collects the following: annual electric bill (in dollars) and home size (in square feet). Output from a regression analysis appears below. Regression equation: Annual bill = 0.55 * Home size + 15 Predictor Coef SE Coef T P Constant Home size What is the 99% confidence interval for the slope of the regression line?
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