Tutor Marked Exercise 4 I. Theory Component Where relevant, keep all your middle work as well as your final answer to a minimum of four decimals, unless otherwise stated. 1. A population consists of 5 units: x = 1, 2, 3, 4, 5. All units are given equal probability of selection and a sample of size 2 is selected with replacement. a. Compute the population mean () and population variance (2) and population standard deviation (). [6 marks] b. Graph a probability histogram of the population values. Title this graph as "Population Distribution." What is the shape of the histogram? [5 marks] c. List all possible samples of size 2 and calculate the mean of each sample. [5 marks] d. Graph a probability histogram of the sample means. Title this graph "Sampling Distribution of Sample Means." What is the shape of the histogram? [5 marks] e. Find the mean, variance, and standard deviation of the sample means. [6 marks] f. Compare your results from (a) and (e). What did you notice? [1 mark] g. Compare the shape of probability histogram in (b) and probability histogram in (d). What do you notice? [1 mark] 2. The average house price in Edmonton for October 2009 was $371, 947 with a standard deviation of $21,000. Let x be the mean price of 49 houses. Find the probability that x : a. is greater than $372,325. [4 marks] b. will fall between $371,638 and $372,256. [5 marks] 3. A random sample of 25 working women yielded a sample income of $50,000. Assume that the standard deviation of all incomes of working women is $5000. a. Find a 95% confidence interval for . [3 marks] b. Find a 99% confidence interval for . [3 marks] c. Refer to (a). Keeping everything else the same, what happens to the width of the confidence interval as the confidence level is decreased? [1 mark] d. Refer to (a). Keeping everything else the same, what happens to width of the confidence interval, if the sample size is quadrupled for the same confidence level? [1 mark] 4. According to a survey, only 40% of regular church goers read their religious books daily. The sample size was 1006. Construct a 90% confidence interval for population proportion of all church goers who read their religious books daily. [4 marks] 5. In each case, find the appropriate sample size required to construct a 95% confidence interval for p (population proportion) that has a margin of error 0.08. a. Assume that no preliminary estimate for p is available. [3 marks] b. Assume that the preliminary estimate for p is 0.35. [3 marks] 6. A manufacturer of packaged juices uses machines on an assembly line to package the juices in 8-ounce plastic packages. Although the packages are labeled as 8 ounces, the company wants the packages to contain a mean fill of 8.17 ounces, so that virtually none of the packages contain less than 8 ounces of juice. In a random sample of 49 juice packages taken off the assembly line, the mean amount dispensed (filled) is 8.159 ounces, with a sample standard deviation of 0.051 ounces. Conduct the appropriate test of hypothesis at the 5% level of significance to determine if the sample evidence supports the view that the population mean amount of juice dispensed (filled) by the machines is different from 8.17 ounces. In your answer make sure you display: H0 and Ha in math terms; a sketch of the sampling distribution showing the critical value(s) and the rejection region(s); compute the test statistic; and also state the decision to reject or not rejectH0; and state the conclusion in the context of the question. [8 marks] 7. During the 2006 season the home team won 136 of the 240 regular season National Football League games. Is this strong evidence of a "home field advantage" in professional football? Note that a "home field advantage" means that the home team wins more than 50% of the games played in a season. Conduct the appropriate test of hypothesis at the 1% level of significance. Keep any calculations you make to 4 decimal places. In your answer make sure you display H0 and Ha in math terms, a sketch of the sampling distribution showing the critical value(s) and the rejection region(s), compute the test statistic, and also state the decision to reject or not rejectH0, and state the conclusion in terms of the context of the question. [8 marks] 8. A mathematics journal recently reported that the mean IQ of all statistics professors is at least 120. To test this claim, a random sample of 16 statistics professors is selected. The sample mean IQ score is computed to be 121, and the sample standard deviation IQ score equals 12. Conduct the appropriate test of hypothesis at the 2% level of significance. In your answer make sure you display H0 and Ha in math terms, a sketch of the sampling distribution showing the critical value(s) and the rejection region(s), compute the test statistic, and also state the decision to reject or not rejectH0, and state the conclusion in terms of the context of the question. [8 marks] II. Computer Component Download the Minitab Project file called TME_Unit4 from the appropriate Assignment Drop Box and save it to your hard drive or memory device. Open the TME_Unit4 Project File. The data required for questions 9, 10, and 11 are listed in the worksheet located in the TME_Unit4 Minitab project file. Where relevant, do NOT round off the results you get from Minitab. Make sure that for each computer problem you paste/display your results to the Minitab Report Pad window AND that you save all your answers to the ONE Minitab project file TME_Unit4. Note: When using Minitab to conduct a test of hypothesis, make sure that you paste the output generated by Minitab to the Report Pad window. Below this output in Report Pad, make sure that you type in the following summary: i. The null and alternate hypothesis ii. The appropriate standardized test statistic along with the related Pvalue (based on the output generated by Minitab) iii. The appropriate decision to either reject the null or fail to reject. iv. Interpret the decision in the context of the original claim 9. A certain brand of apple juice is supposed to have 64 ounces of juice. However, the filling machine is not precise and the exact amount varies from bottle to bottle. The quality control supervisor took a random sample of 24 bottles of juice and measured the contents to verify whether the target mean is met or not. The data is given below: 62.01 63.05 62.39 60.95 63.00 61.03 64.45 63.65 63.72 64.05 66.53 62.08 61.76 65.99 65.05 65.36 67.97 64.08 64.96 63.98 64.93 69.00 67.03 66.95 a. Because the sample size is small, the quality control supervisor must verify that the amount of juice is normally distributed and the sample does not contain any outliers. Verify this using normal probability plot. Paste the normality plot on to the Minitab Report Pad and type in your comments in Report Pad. [2 marks] b. Using Minitab, construct a 99% confidence interval for the mean amount of juice in the juice box. Save your results in the Minitab Project file TME_Unit4. Interpret your answer. [1 mark] c. Use Minitab to test the hypothesis, assuming that = 0.58 at the 0.01 level of significance. Paste the appropriate Minitab output in Report Pad. Below this output type in the appropriate four-step summary in Report Pad, as described above. [4 marks] d. Does the confidence interval from part b agree with your conclusion in part c above? Type in your comments and save your results in the Minitab Project file TME_Unit4. [1 mark] 10.An engineer wants to measure the bias in a pH meter. He uses the meter to measure the pH in 18 neutral substances (pH= 7.00). His data is given below: 7.01 7.05 7.03 7.03 6.97 6.98 6.95 7.01 6.98 7.05 7.02 7.00 7.05 6.99 7.03 6.96 7.01 7.01 a. Because the sample size is small, the engineer must verify that pH is normally distributed and the sample does not contain any outliers. Verify this using a normal probability plot. Save your results in the Minitab Project file TME_Unit4. [2 marks] b. Is there evidence to support the claim that the pH meter is not unbiased. Use Minitab to test the hypothesis, assuming that all required conditions are met, at the = 0.05 level of significance. Paste the appropriate Minitab output in Report Pad. Below this output type in the appropriate four-step summary in Report Pad, as described above. [4 marks] c. Construct a confidence interval with the appropriate level of confidence and verify your conclusion from part (b)? Write your comments and save your results in the Minitab Project file TME_Unit4. [1 mark] 11.A researcher wants to test the claim that at least 40% of the school teachers read at least one book on professional development a year. She contacted a random sample of 36 school teachers and asked whether they had read at least one book the previous year. The responses are given below: Y N Y N N Y N Y Y Y N Y N Y Y Y Y N N N N N N N Y N Y N N Y Y Y Y Y Y Y (Y = Yes, they have read at least 1 book on professional development) (N = No, they haven't read at least 1 book on professional development) a. Is there evidence to support the researcher's claim? Use Minitab to test the hypothesis, assuming that all required conditions are met, at the = 0.10 level of significance. Paste the appropriate Minitab output in Report Pad. Below this output type in the appropriate four-step summary in Report Pad, as described above. [4 marks] b. Construct a 90% two-sided confidence interval for the proportion of teachers who read at least one professional development book a year. Interpret the interval. [1 mark]