Stat 226 Spring 2017 All Sections Homework 5 Due Monday, March 20, 7pm Submission: via Blackboard 1. Confidence Interval Given. Assume I created a 95% confidence interval for the mean hours studied for a test based on a random sample of 100 students. The lower bound of this interval was 6 and the upper bound was 10. Assume that when I created this interval I knew the population standard deviation. Using this information, (a) Calculate the width of the interval. (b) Calculate the margin of error for the interval. (c) Calculate the center of the interval. (d) What is the sample mean? (e) Calculate the population standard deviation. [Round to 3 decimal places.] 2. Critical Values from the t-table (Table D). (a) Accurate to the nearest 3 decimal places, what is the critical value (t or z ) that corresponds to the given confidence levels and degrees of freedom. Fill in the following table with the appropriate critical values (use the rule discussed in class when the exact df are not listed in Table D). Confidence level unknown (t ) known (z ) df = 17 df = 40 df = 75 50% 70% 90% 95% (b) For a given level of confidence C, how does t compare to z ? Note that this holds true for all degrees of freedom values as long as they are finite. i. As long as the degrees of freedom are finite, t will always be less than z . ii. As long as the degrees of freedom are finite, t will always be greater than z . iii. As long as the degrees of freedom are finite, t will always be the same as z . (c) For a given level of confidence C, how does t change compared to z as the df increase? i. As n increases, t gets closer to z . ii. As n increases, t gets farther away from z . iii. As n increases, t stays the same distance away from z . (d) For a given df value, how does t change as C increases? i. For a given df value, t decreases as C increases. ii. For a given df value, t increases as C increases. iii. For a given df value, t stays the same as C increases. 1 3. Netflix Addiction. Researchers are concerned about the impact of Netflix on student's focus and if the amount of time spent watching Netflix tends to poorer academic performance. First, the researchers need to find out, on average, how many hours a week students spend watching or playing a Netflix video. A random sample of 300 students resulted in a sample mean of 10.7 hours watched per week. They know from previous studies that the population standard deviation of this variable is 4.2 hours. (a) [Free Response.] In the context of this problem, what does represent? (b) Calculate a 90% confidence interval, for the unknown population mean, . Round the upper and lower bound values to two decimal places. i. Find the appropriate (positive) critical value using Table A. (Give your answer to 2 decimal places.) ii. What is the value of the lower bound of the 90% confidence interval? (Round your answer to 2 decimal places). iii. What is the value of the upper bound of the 90% confidence interval? (Round your answer to 2 decimal places). (c) [Select All That Apply.] Which of the following represents a correct interpretation of the 90% confidence interval for the mean monthly return rate ? i. There is a 90% chance that the confidence interval calculated in part (b) contains the known sample mean ( x). ii. We are 90% confident that the known sample mean ( x) falls between the bounds calculated in part (b). iii. We are 90% confident that the true (unknown) population mean () falls between the bounds calculated in part (b). iv. There is a 90% chance that the confidence interval calculated in part (b) contains the unknown population mean (). v. If we take 100 samples and calculate 90% confidence intervals for each, 90 out of the 100 intervals will contain the true (unknown) population mean (). (d) One of the researchers is interested in an 74% confidence interval for the unknown population mean, . i. Find the positive z-score (z ) for the corresponding confidence level. (Round your answer to 2 decimal places). ii. Find the lower and the upper bound of the confidence interval. (Round your answer to 2 decimal places). iii. [Free Response.] Suppose that the number of hours that students watch does not follow a Normal distribution. Is the confidence interval trustworthy? That is, do you think you can really be 74% confident? Explain. (State Yes or No first followed by your explanation). 2 4. Coffee Waiting Times. Statistics students were interested in whether students really wait on average 5 minutes from the \"5 minute\" signs at Caribou. They took a random sample of 50 students waiting at the sign in line and recorded the time (in minutes) it took for each student to place their order from that point. The data can be found in the JMP file \"coffee.jmp\". (a) Use JMP to obtain the value of the sample mean x. (Round your answer to the nearest 2 decimal places.) (b) Use JMP to obtain the value of the sample standard deviation s. (Round your answer to the nearest 2 decimal places.) (c) Report the degrees of freedom associated with the t-distribution needed to analyze this data. (d) Based on your answer in (c), what is the value of t for the 95% confidence interval for the mean wait time? (Obtain t from Table-D, accurate to 3 decimal places.) (e) What is the margin of error for the above 95% confidence interval? (Round your answer to the nearest 2 decimal places.) (f) Calculate the 95% confidence interval for the mean wait time in minutes, using your answers from parts (a)-(d). i. What is the value of the lower bound of the 95% confidence interval? (Round your answer to the nearest 3 decimal places). ii. What is the value of the upper bound of the 95% confidence interval? (Round your answer to the nearest 3 decimal places). (g) [Free Response.] Interpret the confidence interval you constructed above in the context of the problem. (h) [Free Response.] Verify your answer in (f) using JMP. To do so follow these steps: Go to Analyze Distribution. Select the column containing the data and click Y, Columns, then click OK. From the output choose the table called Summary Statistics: the upper and lower CI bound correspond to the Upper 95% Mean and Lower 95% Mean. Obtain a screen shot of the this output and upload the screenshot to Blackboard. (Screenshots should be .png or .jpeg files). 5. True or False. (a) All other factors remaining the same, increasing our sample size, n, will decrease the width of our confidence interval. (b) We expect the percentage of 95% confidence intervals for the mean that contain the sample mean to be 95%. (c) The t-distribution is symmetric about 0. (d) The t-distribution approaches the standard normal distribution as the degrees of freedom increase. (e) A 90% confidence interval estimate for a population mean is determined to be 95 to 105. If the confidence level is reduced to 80%, the confidence interval for will be wider than 10 units. 3