# Statistics

Statistics. True or false

1. In an experiment involving matched pairs, a sample of 14 pairs of observations is collected. The degree of freedom for the t statistic is 13.

2. In testing the difference between two means from two independent populations, the sample sizes have to be equal to be able to use the Z statistic.

3. In testing the difference between the means of two independent populations, if neither population is normally distributed, then the sampling distribution of the difference in means will be approximately normal provided that the sum of the sample sizes obtained from the two populations is at least 30. 4. If the limits of the confidence interval of the difference between the means of two normally distributed populations were 0.5 and 2.5 at the 95% confidence level, then we can conclude that we are 95% certain that there is a significant difference between the two population means. 5. When comparing two population means based on independent random samples, the pooled estimate of the variance is used if the population standard deviations are not known and assumed equal.

Chapter 12 11. The chi-square distribution is a continuous probability distribution that is skewed to the left.

12. In a contingency table, when all the expected frequencies equal the observed frequencies the calculated 2 statistic equals 1.

13. In a contingency table, if all of the expected frequencies equal the observed frequencies, then we can conclude that there is a perfect dependence between rows and columns.

14. In performing a chi-square test of independence, as the difference between the respective observed and expected frequencies decrease, the probability of concluding that the row variable is independent of the column variable increases. 15. When we carry out a chi-square test of independence, the expected frequencies are based on the alternative hypothesis.

Multiple Choices 1. If a null hypothesis is rejected at a significance level of .01, it will ______ be not rejected at a significance level of .05  A. Always B. Sometimes C. Never

2. If a null hypothesis is rejected at a significance level of .05, it will ______ be not rejected at a significance level of .01  A. Always B. Sometimes C. Never

4If you live in California, the decision to buy earthquake insurance is an important one. A survey revealed that only 133 of 337 randomly selected residences in one California county were protected by earthquake insurance. Calculate the appropriate test statistic to test the hypotheses that at least 40% buy the insurance.

A. 0.20 B. 0.39 C. -0.13 D. -0.20 E. -0.39

6. If the Z statistic (critical value) is used in lieu of the t statistic when comparing two means from independent populations using small samples, the probability of not rejecting the null hypothesis __________.  A. Increases B. Decreases C. Remains the same D. Depends on whether the population is normal

Essay Type Question (show your work) 1. The average waiting time per customer at a fast food restaurant has been 7.5 minutes. The customer waiting time has a normal distribution. The manager claims that the use of a new cashier system will decrease the average customer waiting time in the store. Based on a random sample of 25 customer transactions the mean waiting time is 6.64 minutes and the standard deviation is 2 minutes per customer. Test the manager’s claim at 5% and 1% significance level tests.

2. In an early study, researchers at an Ivy University found that 33% of the freshmen had received at least one A in their first semester. Administrators are concerned that grade inflation has caused this percentage to increase. In a more recent study, of a random sample of 500 freshmen, 180 had at least one A in their first semester Calculate the appropriate test statistic and test the hypotheses related to the concern and test at 5% and 1%.

3. A microwave manufacturing company has just switched to a new automated production system. Unfortunately, the new machinery has been frequently failing and requiring repairs and service. The company has been able to provide its customers with a completion time of 6 days or less. To analyze whether the completion time has increased, the production manager took a sample of 36 jobs and found that the sample mean completion time was 6.32 days with a sample standard deviation of 1.2 days. At significance levels of 5% and 10%, test whether the completion time has increased. Indicate which test you are performing; show the hypotheses, the test statistic and the critical values and mention whether one-tailed or two-tailed.

4. At  α = 0.05 and 0.10, test the hypothesis that the proportion of Consumer (CON) industry companies winter quarter profit growth is more than 1 percentage point greater than the proportion of Banking (BKG) companies winter quarter profit growth, given that  p(CON)= 0.20,  p(BKG) = 0.15, n(CON )= 380, n(BKG)= 400.

5. The mid-distance running coach, Zdravko Popovich, for the Olympic team of an eastern European country claims that his six-month training program significantly reduces the average time to complete a 1500-meter run. Five mid-distance runners were randomly selected before they were trained with coach Popovich’s six-month training program and their completion time of 1500-meter run was recorded (in minutes). After six months of training under coach Popovich, the same five runners’ 1500 meter run time was recorded again the results are given below. At alpha levels of 0.05 and 0.01, can we conclude that there has been a significant decrease in the mean time per mile?

6. Test the hypotheses H0: μ1  μ2; H1: μ1 > μ2 at α = 0.05, when 1 = 74.5, 2 = 72, s1 = 3.3, s2 = 2.1, n1 = 6, n2 = 6. Assume equal variances. Indicate which test you are performing; show the hypotheses, the test statistic and the critical values and mention whether one-tailed or two-tailed.

7. In the past, of all the students enrolled in “Basic Business Statistics” 10% earned A’s 30% earned B’s, 25% earned C’s, 25% earned D’s and the rest either failed or withdrew from the course. Dr Johnson is a new professor teaching “Basic Business Statistics” for the first time this semester. At the conclusion of the semester, in Dr. Johnson’s class of 60 students, there were 10 A’s, 20 B’s, 20 C’s, 5 D’s and 5 W’s or F’s. Assume that Dr. Johnson’s class constitutes a random sample. Dr Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different than the historical grade distribution. Use α =.05 and .01 and perform a goodness of fit test.

Statistics

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