I. Solve the given problems and SHOW THE SOLUTION COMPUTATION.
1. A supermarket manager has determined that the amount of time customers spend in the supermarket is approximately normally distributed with a mean of 45 minutes and a standard deviation of 6 minutes. Find the probability that:
a. A customer spends less than 48 minutes in the supermarket.
b. Customer spends between 39 and 43 minutes in the supermarket.
2. It has been estimated that 40% of the televisions that the manufacturer makes need repairs in the first three years operation. A new hotel buys 90 televisions form the company. Find the probability that in the first three years operation:
a. Less than 32 of the television needs repairs.
b. Between 38 and 42 of the televisions, inclusive, needs repairs.
3. A manager of a cafeteria wants to estimate the average time customers wait before being served. A sample of 49 customers has an average waiting time of 8.4 minutes with a standard deviation of 3.5 minutes.
a. With 90% confidence, what can the manager conclude about the possible size of his error in using 8.4 minutes to estimate the true average waiting time?
b. Find a 90% confidence interval for the true average customer waiting time. Show your solution.
c. Repeat part (a) using 98% confidence interval.
d. Repeat part (b) using 98% confidence interval.
4. Two supermarket owners each claim that more customers enter their store than the other’s store. Survey was taken and it was found that in 60 days an average of 810 people per day entered the first store with a standard deviation of 40 people. A corresponding survey found that in 80 days an average of 800 people per day entered the second store with a standard deviation of 30 people.
a. State the hypothesis if this is a two-tailed test.
b. Conduct the two-tailed test with α = .05.
c. Test the hypothesis that µ1 – µ2 = 5 against a two-tailed alternative.
d. Explain under what condition
i. A Type I error would be committed.
ii. A Type II error would be committed.
e. Suppose that α was not given and that the null hypothesis is rejected if x < 58 months and otherwise it is accepted. Find the probability of s Type I error.
5. A bank would like to evaluate whether there is difference in the effectiveness in the methods they use to teach their management trainees. For the 10 employees that are taught by method 1 the average score of an evaluation examination was 86 with standard deviation of 6. For 12 employees taught by method 2 the average score was 81 with a standard deviation of 4.
a. State the hypotheses for the two-tailed test in symbols.
b. Conduct the hypothesis test using α = .05.
6. The manufacturers of the deodorant claim that the mean drying time of their product is at most 15 minutes. A sample consisting of 16 cans of the product was used to test the manufacturer’s claim. The experiment yielded a mean drying time of 18 minutes with a standard deviation of 6 minutes. Test the claim at the 5% significant level.