# Statistics Problems

1. If we have a sample size of 100 and the estimate of the population proportion is 0.10, the mean of the sampling distribution of the sample proportion is:

A. 0.0009

B. 0.10

C. 0.03

D. 0.90

E. 0.09

2. When carrying out a large sample test of H0: m =10 vs. Ha: m # 10 by using a p-value, we reject H0 at level of significance level, alpha when the p-value is:

A. Greater than alpha/2

B. Greater than alpha

C. Less than alpha

D. Less than alpha/2

E. Less than Z*alpha

3. A small town has a population of 20,000 people. Among these 1,000 regularly visit a popular local bar. A sample of 100 people who visit the bar is surveyed for their annual expenditures in the bar. It is found that on average each person who regularly visits the bar spends about $1500 per year in the bar with a standard deviation of $120. Construct a 99 percent confidence interval around the mean annual expenditure in the bar.

(show work)

4. The weight of a product is measured in pounds. A sample of 50 units is taken from a batch. The sample yielded the following results: Mean =75 lbs. and Standard Deviation =10 lbs. Calculate a 99 percent confidence interval for µ .

(Show work)

5. 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 16 customer transactions the mean waiting time is 6.3 minutes and the standard deviation is 2 minutes per customer. Test the manager's claim at 5% and 1% significance level tests.

(show work)

6. 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, 185 had at least one A in their first semester Calculate the appropriate test statistic to test the hypotheses related to the concern and test at 5% and 1%.

(show work)

7. 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 6.5 days with a sample standard deviation of 1.5 days. At significance levels of .05 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.

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#### Solution Summary

This solution is comprised of detailed explanation and step-by-step calculation and analysis of the given problems. Supplemented with Interactive EXCEL sheets and text, this solution provides students with a clear perspective of the underlying concepts.

Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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