# Statistics Problems

Teamwork Unit 4

5-41,

Suppose you are sampling from a population with mean µ=1,065 and standard deviation σ=500. The sample size is n=100. What are the expected value and the variance of the sample mean ˉX?

5-44,

When sampling is for a population proportion from a population with actual proportion p=0.5 using a sample of size n=120, what is the standard deviation of our estimator ?P?

5-48,

In problem 5-41, what is the probability that the sample mean will be at least 1,000? Do you need to use the central limit theorem to answer this question? Explain.

5-50,

In problem 5-44, what is the probability that the sample proportion will be at least 0.45?

5-56,

Explain why the sample variance is defined as the sum of squared deviations from the sample mean, divided by n-1 and not by n.?

5-60,

You need to fill in a table of five rows and three columns with numbers. All the row totals and the columns totals are given to you, ant the numbers you fill in must add to these given totals. How many degrees of the freedom do you have?

5-66,

Three independent random samples are collected, and three sample means are computed. The total size of the combined sample is 124. How many degrees of freedom are associated with the deviation from the sample means in the combined data set? Explain.

5-71,

A new kind of alkaline battery is believed to last an average of 25 hrs of continuous use (in a given kind of flashlight).Assume that the population standard deviation is 2hrs. If a random sample of 100 batteries is selected and tested, is it likely that the average battery in the sample will last less than 24 hours of continuous use? Explain.

5-73

Japan's birthrate is believed to be 1.57 per woman. Assume that the population standard deviation is 0.4. If a random sample of 200 women is selected, what is the probability that the sample mean will fall between 1.52 and 1.62?

Homework Unit 4

5-2,

An auditor selected a random sample of 12 accounts from all accounts receivable of a given firm. The amounts of the accounts, in dollars, are as follows: 87.50. 123.10, 45.30, 52.22, 213.00, 155.00, 39.00, 76.05, 49.80, 99.99, 132.00, 102.11. Compute an estimate of the mean amount of all accounts receivable. Give an estimate of the variance of all amounts.

5-8,

Find five random numbers from 0 to 5,600.

5-10,

Suppose you need to sample the concentration of a chemical in a production process that goes in continuously 24 hours per day, 7 days per week. You need to generate a random sample of six observations of the process over a period of one week. Use a computer, a calculator, or a random number table to generate the six observation times ( to the nearest minute)

5-19,

When sampling is from a population with standard deviation σ= 55, using a sample of size n = 150, what is the probability that ˉΧ will be at least 8 units away from the population mean µ?

5-23,

When sampling is done for the proportion of defective items in a large shipment, where the population proportion is 0.18 and the sample size is 200, what is the probability that the sample proportion will be at least 0.20?

5-28,

Shimano mountain bikes are displayed in chic clothing boutiques in Milan, Italy, and the average price for the bike in the city is $700. Suppose that the standard deviation of the bike prices is $100. If a random sample of 60 boutiques is selected, what is the probability that the average price for a Shimano mountain bike in this sample will be between $680 and $720?

5-29,

A quality control analyst wants to estimate the proportion of imperfect jeans in a large warehouse. The analyst plans to select a random sample of 500 pairs of jeans and note the proportion of imperfect pairs. If the actual proportion in the entire warehouse is 0.35 what is the probability that the sample proportion will deviate from the population proportion by more than 0.05?

5-34,

Suppose that you have two biased estimators of the same population parameter. Estimator A has a bias equal to 1/n ( that is, the mean of the estimator is 1/ n unit away from the parameter it estimate), where n is the sample size used. Estimator B has a bias equal to 0.01 (the mean of the estimator is 0.01 unit away from the parameter of interest). Under what conditions is estimator A better than B?

5-36,

Three random samples of sizes, 30, 48 and 32, respectively, are collected, and the three sample means are computed. What is the total number of degrees of freedom for deviation from the means?

5-38

Your bank sends you a summary statement, giving the average amount of all checks you wrote during the month. You have a record of the amounts of 17 out of the 19 checks you wrote during the month. Using this and the information provided by the bank, can you figure out the amounts of the two missing checks? Explain.

See attached files.

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#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score. Formula for the calculation and Interpretations of the results are also included.

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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