Homework Assignment 5

Stat 226 Fall 2022 Homework5 All Sections Submission: via Canvas Instructions: This homework has ONLY one part (Auto-grading). Keep all decimals in your calculations before you get the final answer. 1. Confidence Interval Given. Assume I created a 99% confidence interval for the mean hours studied for a test based on a random sample of 225 students. The lower bound of this interval was 4.5084 and the upper bound was 12.6457. Assume that when I created this interval, I knew the population standard deviation. Give your answer without rounding . Using this information, (a) Calculate the width of the interval. 8.1463 (b) Calculate the margin of error for the interval. Cannot be determined (c) Calculate the center of the interval. 8.57705 (d) What is the sample mean? 8.57705 (e) What is the z ∗ (or z / α 2 ) used? .01/.005 (f) Calculate the population standard deviation. [Do not use the Empirical Rule.][Report upto two 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 from the t-table (table D) . Remember to truncate down the df when the exact value is not listed in the table. Confidence level σ unknown ( t ∗ ) σ known ( z ∗ ) df = 23 df = 55 df = 97 80% 1.319 1.299 1.292 .841 95% 2.069 2.009 1.990 1.645 90% 1.714 1.676 1.664 1.282 99.5% 3.104 2.937 2.887 2.807 (b) For a given level of confidence C, how does z ∗ compare to t ∗ ? Select ALL correct answers. i. For a given level of confidence C, z ∗ is larger than t ∗ for some degrees of freedom and smaller than t ∗ for some degrees of freedom. 1

ii. For a given level of confidence C, z ∗ is always smaller than t ∗ with any degrees of freedom. iii. For a given level of confidence C, t ∗ gets closer to z ∗ while sample size increases. iv. For a given level of confidence C, t ∗ gets further ways from z ∗ while sample size increases. 3. Jethro’s BBQ Food Truck. Jethro’s BBQ is a local chain of restaurants located at central Iowa. Bruce, the owner of the restaurant located at Ames, is thinking about expanding his business using a food truck that would be located during weekdays at the ISU campus. One important aspect for the planning of such project is to determine what would be a reasonable price range for the food. The prices need to be appealing and affordable for college students and at the same time be high enough so the food truck ends up being profitable. The first goal Bruce sets is to determine how much money on average ISU students spend buying lunch during weekdays. Bruce spent a morning asking students all around campus how much money they spend on buying lunch during weekdays. He got an answer from 72 students yielding to an average daily spend of $ 15.55. Bruce found out that five years ago ISU conducted a very complete study on students spending, and the standard deviation back then for daily spend on lunch was $ 3.75. Bruce assumed the standard deviation has not changed. (a) In the context of this problem, what does µ represent? Select ONE correct answer. i. µ represents the average amount of money these 72 ISU students spend buying lunch during weekdays. ii. µ represents the average amount of money ISU students spend buying lunch during weekdays. iii. µ represents the standard deviation of money these 72 ISU students spend buying lunch during weekdays. iv. µ represents the standard deviation of money ISU students spend buying lunch during weekdays. (b) In the context of this problem, would z critical values or t critical values be more appropriate? i. z critical values would suffice in this case since our standard deviation is “known.” ii. t critical values would suffice in this case since our standard deviation is unknown. iii. z critical values would be more appropriate since our sample size is large enough. iv. t critical values would be more appropriate since our sample size is not large enough. (c) Calculate the upper bound for a 96% confidence interval, for the unknown population mean, µ , by answering the following questions below. Round to two decimal places. i. Find the appropriate critical value. .9104 ii. What is the value of the upper bound of the 96% confidence interval? (Round your answer to 2 decimal places). 16.46 (d) Similarly, now calculate a 82% confidence interval for the unknown population mean, µ . 2