#3 1. The fill amount of bottles of a soft drink is normally distributed, with a mean of 1.0 liters and a standard deviation of 0.025 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be
a. between 0.99 and 1.0 liters?
b. below 0.98 liters?
c. greater than 1.01 liters?
d. The probability is 99% that the sample mean amount of soft drink will be at least how much?
e. The probability is 99% that the sample mean amount of soft drink will be between which two values (symmetrically distributed around the mean)?
2. The branch manager of an outlet (Store 1) of a nationwide chain of pet supply stores wants to study characteristics of her customers. In particular, she decides to focus on two variables: the amount of money spent by customers and whether the customers own only one dog, only one cat, or more than one dog and/or cat. The results from a sample of 80 customers are as follows: • Amount of money spent: 𝑋̅ = £25.34, 𝑆 = £8.22 • 47 customers own only a dog. • 26 customers own only a cat. • 7 customers own more than one dog and/or cat.
a. Construct a 95% confidence interval estimate for the population mean amount spent in the pet supply store.
b. Construct a 90% confidence interval estimate for the population proportion of customers who own only a cat. The branch manager of another outlet (Store 2) wishes to conduct a similar survey in his store. The manager does not have access to the information generated by the manager of Store 1. Answer the following questions:
c. What sample size is needed to have 95% confidence of estimating the population mean amount spent in this store to within ±£1.50 if the standard deviation is estimated to be £9?
d. How many customers need to be selected to have 90% confidence of estimating the population proportion of customers who own only a cat to within ±0.045?
e. Based on your answers to (c) and (d), how large a sample should the manager take?
3. An auditor for a government agency is assigned the task of evaluating reimbursement for office visits to physicians paid by Medicare. The audit was conducted on a sample of 100 of the reimbursements, with the following results: • In 12 of the office visits, there was an incorrect amount of reimbursement. • The amount of reimbursement was 𝑋̅ = £73.70, 𝑆 = £30.55.
a. At the 0.05 level of significance, is there evidence that the population mean reimbursement was less than £80?
b. At the 0.05 level of significance, is there evidence that the proportion of incorrect reimbursements in the population was greater than 0.10?
c. Discuss the underlying assumptions of the test used in (a).
d. What is your answer to (a) if the sample mean equals £80?
e. What is your answer to (b) if 14 office visits had incorrect reimbursements? MAT205B/Business Statistics Homework #3
4. Do male and female students study the same amount per week? In 2007, 58 sophomore business students were surveyed at a large university that has more than 1,000 sophomore business students each year. The file StudyTime contains the gender and the number of hours spent studying in a typical week for the sampled students.
a. At the 0.05 level of significance, is there a difference in the variance of the study time for male students and female students?
b. Using the results of (a), which t test is appropriate for comparing the mean study time for male and female students?
c. At the 0.05 level of significance, conduct the test selected in (b).
d. Write a short summary of your findings.
5. The per-store daily customer count (i.e., the mean number of customers in a store in one day) for a nationwide convenience store chain that operates nearly 10,000 stores has been steady, at 900, for some time. To increase the customer count, the chain is considering cutting prices for coffee beverages. The question to be determined is how much to cut prices to increase the daily customer count without reducing the gross margin on coffee sales too much. You decide to carry out an experiment in a sample of 24 stores where customer counts have been running almost exactly at the national average of 900. In 6 of the stores, the price of a small coffee will now be $0.59, in 6 stores the price of a small coffee will now be $0.69, in 6 stores, the price of a small coffee will now be $0.79, and in 6 stores, the price of a small coffee will now be $0.89. After four weeks of selling the coffee at the new price, the daily customer count in the stores was recorded and stored in CoffeeSales.
a. At the 0.05 level of significance, is there evidence of a difference in the daily customer count based on the price of a small coffee?
b. If appropriate, determine which prices differ in daily customer counts.
c. At the 0.05 level of significance, is there evidence of a difference in the variation in daily customer count among the different prices?
d. What effect does your result in (c) have on the validity of the results in (a) and (b)?