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Hypothesis Testing & Regression Analysis

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Attached (2):
7 statistic questions (.doc)
Formulas and Tables (for reference)


Problem 1 thru 4 (don't forget to state your hypotheses, type of test, alpha level, and your decision statistic; draw a bell curve diagram indicating rejection region)

PROBLEM 1: Conduct a one-tailed hypothesis test given the information below.

A certain brand of Green Energy light bulbs was advertised as having an average illumination life-span of 2,500 hours. A random sample of 50 bulbs burned out with a mean life-span of 2,470 hours and a sample standard deviation of 140 hours. With a 0.05 level of significance, is the sample mean less than the advertised mean?

PROBLEM 2: Given the following data from two independent samples, conduct a one-tailed hypothesis test to determine if the first sample mean is larger than the second sample mean, given a 0.05 level of significance.

n1 = 55 n2 = 46
xbar1= 72 xbar2 = 62
s1=15 s2 = 12

PROBLEM 3. Conduct a one-tailed hypothesis test given the information below.

A test was conducted to determine whether gender of a spokesperson affected the likelihood that consumers would prefer a new vacuum cleaner. A survey of consumers at a trade show employing a female spokesperson determined that 78 out of 200 customers preferred the product, while 56 of 180 customers preferred the product when a male spokesperson was employed. At the 0.05 level of significance, do the samples provide sufficient evidence to indicate that on the average, fewer consumers prefer a new product when the spokesperson is male?

PROBLEM 4. Conduct a two-tailed hypothesis test given the information below.

Assuming that the population variances are equal for male and female Grade Point Averages (GPAs), use the following sample data to test whether the averages are different at the 0.05 level of significance.

Male GPA's Female GPA's
Sample Size 17 15
Sample Mean 3.8 3.95
Sample Standard Dev .5 .7

PROBLEM 5. Regression Analysis (a-e)

A real estate investor has devised a model to estimate home prices in a new suburban development. Data for a random sample of 30 homes were gathered on the selling price of the home (in units of $1,000), the home size (square feet), the lot size (in units of 1,000 square feet), and the number of bedrooms.

The following multiple regression output was generated:

Regression Statistics
Multiple R 0.9647
R Square 0.9307
Adjusted R Square 0.9227
Standard Error 26.0389
Observations 30

Coefficients Standard Error t Stat P-value
Intercept -34.6165 38.3735 -0.9021 0.3753
X1 (Square Feet) 0.1532 0.0184 8.3122 0.0000
X2 (Lot Size) 9.0024 1.7120 5.2583 0.0002
X3 (Bedrooms) 17.3903 6.8905 2.5238 0.1259

a. Why is the coefficient for lot size a positive number?

b. Which is the most statistically significant variable? What evidence shows this?

c. Which is the least statistically significant variable? What evidence shows this?

d. For a 0.05 level of significance, should any variable be dropped from this model? Why or why not?

e. Predict the sales price of a 2134-square-foot home with a lot size of 13,400 square feet and three bedrooms.

Problem 6: Hypothesis Testing

True/False Questions

a. If the null hypothesis states that there is no difference between the mean net income of retail stores in Chicago and New York City, then the test is two-tailed.

True False

b. If we are testing for the difference between two population proportions, it is assumed that the two samples are large enough that the binomial distribution can be approximated by the normal distribution.

True False

c.If we are testing for the difference between two population means and assume that the two populations have equal and unknown standard deviations, the standard deviations are pooled to compute the best estimated variance.

True False

Multiple Choice Questions

d. If the decision is to reject the null hypothesis of no difference between two population proportions at the 5% level of significance, what are the alternate hypothesis and rejection region?

Problem 7: Define multicollinearity in the following terms:

a. In which type of regression is it likely to occur?

b. What is the negative impact of multicollinearity in a regression?

c. Which method is used to determine if it exists?

d. If multicollinearity is found in a regression, how is it eliminated?

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Solution Summary

The solution provides step by step method for the calculation of testing of hypothesis and regression analysis. The solution also provides a brief information on multicollinearity, its negative impact in regression and its elimination from regression. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

See Also This Related BrainMass Solution

Artsy Company: Difference in pay rates by job grade, time in grade and by sex

Artsy Case
The Artsy Corporation has been sued in the United States Federal Court on charges of employment discrimination under Title VII of the Civil Rights Act of 1964. (Artsy is an actual corporation and the data given in the case is real, but the name has been changed to protect the firm's true identity.) The litigation at contention here is a "class action" lawsuit brought on behalf of all females whom the company employed, or who had applied for work with the company, between 1979 and 1987. Artsy operates in several states, runs four quite distinct businesses, and has many different types of employees. The allegations against Artsy include issues of hiring, pay, promotions, and other "conditions of employment."
In such large class action employment discrimination lawsuits statistical evidence commonly plays a central role in the determination of guilt or damages. In an interesting twist on traditional legal procedures, the precedent in these cases is that plaintiffs may make a "prima-facie" case purely in terms of circumstantial statistical evidence. If that statistical evidence is reasonably strong, the burden of proof shifts to the defendants to rebut the plaintiff's statistics with other statistical data, other statistical analyses of the same data, or by non-statistical testimony. In practice, statistical arguments often dominate the proceedings of such EEO cases. Indeed, in this case the statistical data used filled numerous computer tapes and the supporting statistical analysis comprised thousands of pages of computer printouts and reports. We work here with a small subset of the voluminous data that pertain to one of the several contested issues in one of the company's locations.

Specifically, the data in Table 1 relate to the pay of 256 employees on the bi-weekly payroll at one of the Artsy Company's Pocahontas, Maine production facilities. The data include:
? an identification number (IDNUMBER) that would permit us to identify the person by name or social security number,
? the person's sex (SEX) where a 0 denotes female and a 1 denotes a male,
? the person's job grade in 1986 (GRADE),
? the length of time (in years) the person had been in that job grade as of 12/31/86 (TING), and
? the person's weekly pay rate as of 12/31/86 (RATE). The issue of concern is fair pay for female employees.
The plaintiff's attorneys have proposed settling the pay issues for this group of female employees for a "back pay" lump payment of 25% of their pay during the period 1979 to 1987. It is our task to examine the data in the table for evidence in favor of, or against the charges of pay discrimination against the females. To make our mission explicit suppose that we are to advise the lawyers for the Artsy Company on how to proceed. (An alternative mission would be to assist the plaintiffs.)

Please consider the following issues:
1) Overall, how different is pay by sex? Are the differences in pay statistically significant? Is a statistical hypothesis test appropriate in an issue like this? If so, how should it be done? How could it be explained to a judge? What arguments do you anticipate the plaintiffs will be making with these data?
2) The Artsy Company wishes to argue that a legitimate explanation of any pay rate difference is the difference in job grades by sex. (In this analysis we will tacitly assume that each person's job grade is, in fact, appropriate for them, even though the plaintiff's attorneys have charged that females have been unfairly kept in the lower grades. Other statistical data, not available here, are used in the analysis of the job placement issue.) The company's lawyers ask, "Is there a relatively easy way to understand, to analyze and display the pay differences by job grade? Easy enough that it could be presented to an average jury without confusing them?" Again, try to anticipate the possible arguments of the plaintiffs. To what extent does job grade appear to explain the pay rate differences between the sexes? Propose and carry out appropriate hypothesis tests or confidence intervals to check whether the difference in pay between sexes is statistically significant within each of the grades.
3) In the actual case, the analysis carried out in (2) above suggested to the attorneys that differences in pay rates are due, at least in part, to differences in job grades. They had heard that in another EEO case the dependence of pay rate on job grade had been investigated with regression analysis. Perform a simple linear regression of pay rate on job grade. Interpret the results fully. Is the regression significant? How much of the variability in pay does job grade account for? What light does this analysis shed on the pay fairness issue? Does it help or hurt the Artsy company?
4) It is argued that seniority within a job grade should be taken into account since the Artsy Company's written pay policy explicitly calls for the consideration of this factor. How different are times in grade by sex? Enough to matter?
5) The Artsy legal team wants an analysis of the simultaneous influence of grade and time in grade on pay. Perform a multiple linear regression of pay rate versus grade and time in grade. Is the regression significant? How much of the variability in pay rates does this model explain? Will this analysis help your clients? Could the plaintiffs effectively attack it? Utilize residuals in your analysis of these issues.
6) The attorneys ask: "Is it possible to do a regression analysis that simultaneously considers the effect on pay of grade, time-in-grade and sex?" If so, carry one out.
7) Organize your analyses and conclusions in a brief report summarizing your findings for your client, the Artsy Corporation. Be complete but succinct. Be sure to advise them on the issue of the settlement. Please be as forceful as you can be in arguing "the Artsy Case" without misusing the data or statistical theory. Apprise your client of the risks they face by developing the most forceful counter argument that you believe the female plaintiffs could fairly make.

See attached case file.

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