1. Consider the following sample of nine wait timing measured in seconds at a drive through coffee shop. The population mean and standard deviation are 100 and 20 respectively.
|
125
|
95
|
66
|
116
|
99
|
91
|
102
|
51
|
110
|
a. Calculate the sample standard deviation given the sample mean is 95.
b. What is the size of the sampling error in this case?
c. What is the probability that the average waiting time would be between 90 and 95 seconds?
d. What is the probability that a customer waits more than 65 seconds?
2. An order office receives 20 faxed orders every hour.
a. What is the probability that it will receive 8 orders in the next 15 minutes?
b. What is the probability that an order will be faxed within the next 9 minutes? (HINT: You are given lambda in hours but here it is a question of minutes. You must put them both into the same units of time.)
c. What is the probability that more than 12 minutes will elapse between faxed orders?
3. The student body of a local college draws 40% of its students from Ontario, 35% from the rest of Canada, and the remainder from outside Canada. Of those from Ontario, 60% are female, of those from the rest of Canada 25% are female, and of those from outside Canada, 15% are female.
a. What is the probability of a male student from Ontario?
b. What is the probability of a female student?
c. What is the probability of a male student who did not come from Ontario?
4. The number of cups of coffee served at a local Tim Horton's during the morning rush hour between 7 and 9 AM is normally distributed. Less than 200 cups served is considered a slow day, 200- 250 an average day, and over 250 a busy day. A random sample of five days was taken and the sales figures observed. The results are as follows:
a. Calculate the sample average and sample standard deviation for this data.
b. Construct a 99% confidence interval for the population average sales for this period.
c. Management is concerned that the average is falling below the national average of 210 cups during this period. Test this hypothesis using alpha = 0.05.
d. What assumptions must hold to allow you to construct the confidence interval and test the hypothesis in this case?
5. A sample in 2003 found that 4.8% of a total of 250 Canadians had been victims of identity theft.
a. Construct a 98% confidence interval for the true proportion of Canadians who have been victims of identity theft.
b. A research study proposed that more than 3% of Canadians are victims of this crime. Test this hypothesis for alpha = 0.01 and the sample data from 2003.
6. The life insurance industry maintains that the average worker in Saskatoon has less than $25,000 of personal life insurance. You believe it is higher. You sample 100 workers in Saskatoon at random and find the sample average to be $26,650, The population standard deviation for this type of insurance is known to be $12,000. Use α=0.05 throughout.
a. Test your belief using a significance level of 5%.
b. Explain, in the context of this question, what is meant by a Type I error, a Type II error, and the power of the test?
c. If the true average for this population is in fact $30,000, what is the probability of committing a Type II error?
d. Calculate the power of the test.
7. Shoplifting costs retail businesses a great deal of money every year. In spite of this the historical evidence suggests that only 50% of all shoplifters are turned over to the police. A random survey of 40 retailers revealed that 24 of them had turned their most recent shoplifter overto the police. Use alpha = 0.05 for all tests.
a. Test to see if this sample data indicates that more shoplifters are being turned in than in the past?
b. What assumptions must hold for this test to be performed?
c. Find the p-value for part a. (It is the probability that z is greater than the absolute value of the test statistic in part a.)
d. Describe both Type I and Type II errors in the context of this problem.
e. Calculate the probability of Type II error if the true proportion of shoplifters turned over to police now is 55%.
f. (5) Calculate the probability of Type II error if the true proportion of shoplifters turned over to police now is 55%,if the sample size is increaed to 100.
g. What does this tell you about the relationship between alpha, beta and sample size?