You are the Director of Human Resources for a Gym.You want to increase the usage of the facility among the existing members belonging to a category i.e. working in jobs which require high fitness standards (e.g. models, pilots, industry plant workers etc). You want to gather some inferential statistics about your gym member's income.You also wish to determine if such gym member's usage of the facility per week is useful for predicting the income (in $) of gym member.

THE DATA:The Gym usage per week of 180 randomly selected members are contained in the file: Fitness.xlsx, located in Week 11.For linear regression, the X-variable is the Gym usage per week, and the Y-variable is Income.

INSTRUCTIONS:Answer all the questions below. All calculations must be performed with Excel or PHStat. Attach Excel or PHStat output where indicated.You will receive zero credit for any answer lacking the required Excel or PHStat output.

ROUND OFF ALL CALCULATIONS TO AT LEAST FOUR DECIMAL PLACES.Highlight the cells with output where decimal places need setting.Then use the "Increase Decimal" tool on Excel's Home/Number menu to four decimal places.If you have problems obtaining the required decimal places, contact me.

1.Find the mean and standard deviation of the Income:[4 POINTS]

PASTE EXCEL DESCRIPTIVE STATISTICS BELOW.

2.Assume that the population is normally distributed, but the population standard deviation is not known.Use your sample data to find a 95% confidence interval for the true mean income of all such members in the gym: [6 POINTS]

PASTE PHSTAT OUTPUT BELOW:

State the margin of error of the confidence interval:

3.Assume that the population standard deviation is 0.30 and assume the population is approximately normally distributed.Find the sample size that would be required to determine a 95% confidence interval for the true mean income of gym members if we want to be within 0.10 of the true mean. That is, we want the margin of error, e, to not exceed 0.10.[4 POINTS]

PASTE PHSTAT OUTPUT BELOW:

4.Using your sample data, test this hypothesis at the alpha = 0.01 significance level.You may assume that the population standard deviation is not known and that the population is approximately normally distributed.[14 POINTS]

(a)Is there sufficient evidence to conclude that the mean income for all such gym members is more than 90000?

PASTE PHSTAT OUTPUT BELOW:

(b)Test the hypothesis again, changing alpha to 0.05 but not changing anything else.

PASTE PHSTAT OUTPUT BELOW:

Now mark all of the following statements about the two hypothesis tests either T(TRUE) or F(FALSE).

__F_ The p-value is the probability that the null hypothesis will be rejected.

_T__ The second test has a smaller "reject" region than the first.

___T__The test statistic measures the distance between the mean being tested and the sample mean.

__F_ The null hypothesis will be rejected provided alpha exceeds the p-value.

___T___The critical value is the boundary between the "reject" region and the "do not reject" region.

___F___The p-value is the probability of getting a test statistic equal to, or more extreme than the sample result, if the null hypothesis is true.

5.Suppose it is known that 50 out of the 180 employees in the sample are women.[8 POINTS]

(a)Find a 95% confidence interval for the true proportion of all such Gym members who are women.

PASTE PHSTAT OUTPUT BELOW:

(b)What is your opinion of the precision of this confidence interval?Give a reason for your answer.

6.Assume that the population proportion is 0.45, and find the sample size that would be required to determine a 95% confidence interval if we want to be within 0.05 of the true proportion of such women gym members. That is, we want the margin of error, e, to not exceed 0.05.[4 POINTS]

PASTE PHSTAT OUTPUT BELOW:

LINEAR REGRESSION - Use the sample to complete the following section.Remember, the X variable is Gym usage per week, and the Y variable is Income

7.PASTE A SCATTER PLOT BELOW:[4 POINTS]

8.Perform the regression analysis using PHSTAT and PASTE THE PRINTOUT BELOW:[4 POINTS]

NOTE:BEFORE YOU COPY THE PRINTOUT, CHANGE THE FORMAT OF THE P-VALUE FOR GYM USAGE PER WEEK TO SCIENTIFIC NOTATION.HIGHLIGHT THE CELL, THEN ON THE EXCEL HOME/NUMBER MENU, SELECT "SCIENTIFIC" FROM THE DOP-DOWN BOX.

9. The regression output. [10 POINTS]

i.The regression equation is:__________

ii.The slope of the equation is:___________________________________

iii.The y-intercept of the equation is:_________________________

iv.The standard error of the estimate is:

v.The coefficient of determination is:

10. Using the Excel printout from Question 8, test the hypothesis that there is no linear relationship between X and Y. Test at alpha = 0.05 significance level.[8 POINTS]

i.State the null hypothesis:_____ __________________

ii.State the alternate hypothesis:___ ________________

iii.p-value: _______

iv.Test result and reason for test result:________ ___________________________

11.Interpretation. [6 POINTS]

(a)What does the y-intercept of this regression equation represent?

(b)State the exact meaning of the slope in this regression equation.

(c)Predict the salary of such a member with a gym usage per week of 6 days

12. [12 POINTS](a)PASTE RESIDUAL PLOT BELOW:

(b)From the residual plot, do you think that the two regression assumptions listed below are satisfied?Give the reason for your conclusion.

Linearity:_ __________________________________

Reason:__ ______________________________

Equal Variance:____ __________________________

Reason:_________ __________________________

13. [8.POINTS]

(a)PASTE A NORMAL PROBABILITY PLOT OF RESIDUALS BELOW:

(b)From the normal probability plot, do you think the normality assumption for regression is satisfied?Give the reason for your conclusion.

Yes, approximately linear.

14. Determine 95% confidence and prediction intervals for X = 7.[4 POINTS]

PASTE PHSTAT OUTPUT BELOW:

15. Discuss this model.How good do you think the model is for predicting Income of the category of gym members whose jobs require high fitness levels?Give reasons for your answer.Then state at least two other possible independent variables that you think would be useful for predic