# Problem 1. Suppose we are given two independent random samples of s...

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Problem 1. Suppose we are given two independent random samples of sizes n and from Bernoulli populations with parameters P1 and P2- Let P1 and P2 be the corresponding sample proportions. Consider the problem of testing the hypotheses Ho:p P2 0 versus H2:P1-P2=0 (a) (1 point) Read Section 9.2.1 from the book. and write down the test statistic for the above hypothesis (b) Now, suppose random sample of 220 female and 210 male coffee drinkers was selected and interviewed. The result was that 71 women and 58 men indicated a preference for decaffeinated coffee. (i) (2 points) Should we conclude at 5% significance level, that the proportion of female coffee drinkers who prefer decaffeinated coffee differs from the proportion of male coffee drinkers who prefer decaffeinated coffee? (ii) (2 points) Construct 95% confidence interval for the difference in proportions between male and female coffee drinkers who prefer decaffeinated coffee. Problem 3. (5 points) To study the effectiveness of certain commercial liquid protein diet, the Food and Drug Administration sampled nine individuals who were entering two-week weight loss program. Their weights immediately before and 6 months after completing the program are recorded below: Person Weight before Weight after 1 197 185 2 212 220 3 188 180 4 226 217 5 170 185 6 194 197 7 233 219 8 166 170 9 205 202 Based on the data, would you conclude that the diet is effective for weight loss? Use 5% significance level. What assumptions have you made? Problem 4. (5 points) A study was instigated to see if southern California earthquakes of at least moderate size (having values of at least 4.4 on the Richter scale) are more likely to occur on certain days of the week than on others. The following data were obtained for 1100 earthquakes: Day Sun Mon Tues Weds Thurs Fri Sat Number of Earthquakes 156 144 170 158 172 148 152 (a) Test the hypothesis that an earthquake is equally likely to occur on any of the seven days of the week. Use 5% significance level. (b) What is the p-value of the data? Problem 6. (a) (3 points) A random sample of 187 voters is chosen. and the voters are asked to evaluate the performance of the first 100 days of the US president. Use the resulting data to test the hypothesis that the evaluation of an individual does not depend on whether that individual is man or woman. Use 10% level of significance. Women Men Positive evaluation 54 47 Negative evaluation 20 32 Not sure 23 11 (b) (1 point) Now repeat the exercise in (a). after doubling all the count data. (c) (1 point) Compare the p-values in (a) and (b). If your answers are different provide an intuitive explanation for the difference. Problem 8. An electric utility company wants to estimate the relationship between the daily summer temperature and the amount of electricity used by its customers The data is given in the table below: (a) (2 points) Construct scatter plot for the data. (b) (2 points) What is the equation of the least squares regression line? Draw the least squares line on the seatter plot in (a), and compute the R² value for the linear fit. (c) (2 points) Construct a residual plot and a normal QQ-plot of the residuals, and comment (qualitatively) on the appropriateness of the linear model assumption (d) (2 points) Suppose the temperature for tomorrow is predicted to be 93°F Provide a point estimate for the amount of electricity that will be consumed tomorrow. (e) (2 points) Test the hypothesis, at 5% significance level, that the daily temperature has no effect on the amount of electricity consumed. (Assume the date follow linear model.) Temp (F) Electricity (millions of kW) 85 22.5 90 23.7 76 20.3 91 23.4 84 24.2 94 23.5 88 22.9 85 22.4 97 26.1 86 23.1 82 22.5 78 20.9 77 21.0 83 22.6

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