1. The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 350 hours.
- At the 0.05 level of significance, is there evidence that the mean life is different from 375 hours?
- Compute the p-value and interpret its meaning.
- Construct a 95% confidence interval estimate of the population mean life of the light bulbs.
- Compare the results of (a) and (c). What conclusions do you reach?
2. Suppose that in Problem 1, the standard deviation is 120 hours.
- Repeat (a) through (d) of Problem 1 assuming a standard deviation of 120 hours.
- Compare the results of (a) to those of Problem 1.
3. The manager of a paint supply store wants to determine whether the mean amount of paint contained in 1-gallon cans purchased from a nationally known manufacturer is actually 1 gallon. You know from manufacturer's specifications that the standard deviation of the amount of paint is 0.02 gallon. You select a random sample of 50 cans, and the mean amount of paint per 1-gallon can is 0.995 gallon.
a. Is there evidence that the mean amount is different from 1.0 gallon (use alpha =0.01)?
b. Compute the p-value and interpret its meaning.
c. Construct a 99% confidence interval estimate of the population mean amount of paint.
d. Compare the results of (a) and (c). What conclusions do you reach?
4. Suppose that in Problem 3, the standard deviation is 0.012 gallon.
a. Repeat (a) through (d) of Problem 3, assuming a standard deviation of 0.012 gallon.
b. Compare the results of (a) to those of Problem 3
5. You are the manager of a restaurant for a fast-food franchise. Last month, the mean waiting time at the drive-through window for branches in your geographical region, as measured from the time a customer places an order until the time the customer receives the order, was 3.7 minutes. You select a random sample of 64 orders. The sample mean waiting time is 3.57 minutes, with a sample standard deviation of 0.8 minute
a. At the 0.05 level of significance, is there evidence that the population mean waiting time is different from 3.7 minutes?
b. Because the sample size is 64, do you need to be concerned about the shape of the population distribution when conducting the t test in (a)? Explain.
6. A stationery store wants to estimate the mean retail value of greeting cards that it has in its inventory. A random sample of 100 greeting cards indicates a mean value of $2.55 and a standard deviation of $0.44.
a. Is there evidence that the population mean retail value of the greeting cards is different from $2.50? (Use a 0.05 level of significance.)
b. Determine the p-value and interpret its meaning
7. In a recent year, the Federal Communications Commission reported that the mean wait for repairs for Verizon customers was 36.5 hours. In an effort to improve this service, suppose that a new repair service process was developed. This new process, used for a sample of 100 repairs, resulted in a sample mean of 34.5 hours and a sample standard deviation of 11.7 hours.
a. Is there evidence that the population mean amount is less than 36.5 hours? (Use a 0.05 level of significance.)
b. Determine the p-value and interpret its meaning
8. 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 by approximately half. The small size will now be $0.59 instead of $0.99, and the medium size will be $0.69 instead of $1.19. Even with this reduction in price, the chain will have a 40% gross margin on coffee. To test the new initiative, the chain has reduced coffee prices in a sample of 34 stores, where customer counts have been running almost exactly at the national average of 900. After four weeks, the sample stores stabilize at a mean customer count of 974 and a standard deviation of 96. This increase seems like a substantial amount to you, but it also seems like a pretty small sample. Do you think reducing coffee prices is a good strategy for increasing the mean customer count?
a. State the null and alternative hypotheses.
b. Explain the meaning of the Type I and Type II errors in the context of this scenario.
c. At the 0.01 level of significance, is there evidence that reducing coffee prices is a good strategy for increasing the mean customer count?
d. Interpret the meaning of the p-value in (c).