# Regression analysis and T test.

1. The following information should be used in conjunction with questions 1 to 6.

Dave's Diners, a chain of cafes, is planning to open another one. Dave (the owner) believes that the size of the student population at nearby university campuses is related to sales revenue in the cafes. Dave and his management team collect data from a sample of 10 of its diners located near college campuses. The table below summarizes their findings.

Student

Population 2 6 8 8 12 16 20 20 22 26

(1000s)

Annual

Sales 58 105 88 118 117 137 157 169 149 202

(Â£1000s)

Which option gives the least squares fit straight line to this data so that it may be used to predict sales from the size of the student population?

2. Which option gives the percentage of variation in sales explained by student population size?

3. Using the line found in question 1 which option gives the predicted annual sales of a cafÃ© with a student population of 11,000?

4. Which option gives the 95% confidence interval for the average annual sales with a student population of 11,000 students?

5. If the student population were to increase by 1000, which option gives the resulting average increase in annual sales?

6. Which option gives a 95% confidence interval for the increase in average annual sales found in question 5?

7. The following information should be used in conjunction with questions 7 to 13.

A soup company markets a number of varieties of 'homemade' soup. The standard-size soup can holds a maximum of 11 ounces., while the label on each can advertises contents of 10.5 ounces. The extra 0.5 ounce is to allow for the possibility of the automatic filling machine placing more soup than the company actually wants in a can. Past experience shows that the number of ounces placed in a can is approximately normally distributed , with a mean of 10.65 ounces and a standard deviation of 0.1 ounce. Which option gives the proportion of cans which overflow?

8. Which option gives the probability that a can has less soup than advertised?

9. If the chance of a can having less soup than advertised is to be no more than 1%, which option gives the minimum mean fill that the automatic filling machine can be set?

10. The following information should be used with questions 10 to 13.

A random sample of 10 cans is selected from the filling process and their contents examined.

The following are the individual contents in units of ounces.

10.54 10.62 10.47 10.33 10.60 10.71 10.83 10.34 10.77 10.48

Which option gives the mean fill of these cans?

11. Which option gives the standard deviation of the fill of the 10 cans?

12. Which option gives a 95% confidence interval for the mean fill of the cans?

13. If a test the hypothesis that the mean fill is not 10.65 ounces is made using the answer to question 13, which option gives the most appropriate answer?

14. The following information should be used in conjunction with questions 14 to 22.

Two assembly lines are designed on the assumption that there should be no difference between mean assembly times of a particular household appliance. Independent tests for the two assembly operations show the following results.

Line 1 Line 2

n1 = 10 n2 = 15

1 x =14.8 minutes 2 x =14.0 minutes

s1=0.8 minutes s2=0.6 minutes

The hypothesis that there is no difference between the mean assembly times for the two processes is going to be tested at the 0.02 significance level.

Which option gives the appropriate test procedure?

15. Which option gives the value of the number of degrees of freedom for this test?

16. Which option gives the observed value of the test statistic?

17. Which option gives the most appropriate conclusion for this test?

18. The following information should be used in conjunction with questions 18 to 23.

The leader of the workers union argues that there is a difference between the two lines, and that operators on Line 1 should be compensated for the extra time it takes to assemble the household appliance. Management argue that there is no difference in the lines and any perceived difference is due to the fact that some operators are faster than others. To see if there is any truth in what the union and management are saying, the management decide to set up an experiment.

A group of 10 workers are randomly selected (five from each line) and placed at random on either Line 1 or on Line 2. After performing the operations each of the 10 workers are moved across from the line they were placed on initially to the other one. The two sets of timings (in minutes) for the 10 workers are as follows:

Worker 1 2 3 4 5 6 7 8 9 10

Line 1 13.6 10.1 11.0 14.9 12.7 10.8 15.1 9.9 9.9 15.3

Line 2 13.5 9.9 10.9 15.0 11.5 10.8 14.7 10.2 9.6 14.8

Which option gives the observed value of the most appropriate test statistic if the hypothesis

that there is no difference between the two assembly lines is being considered?

19. Which option gives the average difference between the two sets of timings?

20. Which option gives the standard deviation of the differences between the two timings?

21. Which option gives a 95% confidence interval for the mean difference between the two sets of timings?

22. Whose argument is supported by your results, management or union? Which option gives the most appropriate answer?

23. From the option list provided select two examples of nominal data.

weight, names, position in race, IQ, temperature, height, colours, preference score,

24. From the option list provided select two examples of ordinal data.

weight, names, position in race, IQ, temperature, height, colours, preference scores

25. From the option list provided select two examples of interval scale data.

weight, names, position in race, IQ, temperature, height, colours, preference score,

26. From the option list provided select two examples of ratio scale data.

weight, names, position in race, IQ, temperature, height, colours, preference score,

27. The following information should be used with questions 27 to 31.

A employment agency provided the following data as an example of selection among 40 male and 40 female applicants for 12 positions.

Applicant Selected Not Selected Total

Male 7 33 40

Female 5 35 40

The chi-square test of independence was suggested as a way of determining if the

decision to hire seven males and five females should be interpreted as having a selection bias in favour of males. Let the population proportion of male and female applicants selected be denoted M π and F π respectively.

Which option gives the null and alternative hypotheses for this chi-square test?

28. How many degrees of freedom are there for this chi-square test?

28. Which option gives the critical chi-square value if we use a 10%significance value?

29. Which option gives the observed value of the chi-square statistic for the above test?

30. Which option gives the most appropriate conclusions of your hypothesis test?

31. Which option gives a 95% confidence interval for the difference between the proportion of males and females selected.

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#### Solution Summary

The solution gives the complete details regression analysis and testing of hypothesis problems. Null Hypothesis, alternative hypothesis, significance level and critical value are given with interpretations. The step by step procedure for the computation of regression equation, regression coefficients and prediction for future values are given for the regression analysis problems.