Question

Owners of Mighty Muscles, Inc., a health club in Hoboken, New Jersey, are trying to increase membership at the club. After reviewing operations at other health clubs in the area, Harry Hunk, the manager at Mighty Muscles, feels that there might be a relationship between the number of members in a health club and several explanatory variables, including (1) monthly dues, (2) local population, and (3) type of weight training equipment in the club. Three kinds of weight machines are commonly found in health clubs: Muscle UP (MU), Iron Man (IM), and Studly Dude (SD). Mr. Hunk collects data for these variables for 20 health clubs similar to Mighty Muscles.

Test whether or not monthly dues are inversely related to the number of members in the club. Use alpha = 0.05.

Analyze the data and determine how well the model fits the data. Interpret your findings and recommend to Harry the policy decisions he should make for the club.

Members	Dues	Pop (000s)	Equip Type
345	22.5	39.7	MU
387	21	43.3	MU
412	20	47.7	MU
434	20	48.9	SD
467	18.5	54.5	SD
490	18	55.7	SD
612	15	62.1	SD
523	17.5	56.7	IM
545	17	58.9	IM
567	16.5	60	IM
578	16	61.2	SD
599	15	63.6	SD
634	14.5	60	IM
656	13	61	SD
669	13.1	61.4	MU
134	30	23.5	MU
243	27	27.8	IM
259	25.5	32.1	IM
278	23	37.3	SD
321	24	35.8	SD

Answer 1

Solution:

To test whether or not monthly dues are inversely related to the number of members in the club using . We have to calculate the correlation coefficient between these two and test its significance using the following formulas:

And the result is given below:

Correlations
	Members	Dues
Members	Pearson Correlation	1	-.990**
Sig. (2-tailed)		.000
Dues	Pearson Correlation	-.990**	1
Sig. (2-tailed)	.000
**. Correlation is significant at the 0.01 level (2-tailed).

This correlation indicates that there is almost perfect linear association between Members and Dues. Also, the correlation matrix indicates that it is significant at both.

To build the regression model we have used all the predictors available and found that the overall ANOVA is highly significant.

ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	452315.255	3	150771.752	316.030	.000b
Residual	7633.295	16	477.081
Total	459948.550	19
a. Dependent Variable: Members
b. Predictors: (Constant), Dues, Equip_Type, Pop

The estimated coefficients are tabulated below:

Coefficientsa
Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.	95.0% Confidence Interval for B
B	Std. Error	Beta	Lower Bound	Upper Bound
1	(Constant)	898.399	180.085		4.989	.000	516.636	1280.161
Pop	1.509	1.842	.124	.819	.425	-2.396	5.413
Equip_Type	10.700	6.924	.052	1.545	.142	-3.979	25.379
Dues	-27.851	4.824	-.864	-5.773	.000	-38.078	-17.624
a. Dependent Variable: Members

The predicted model is as below:

Members (y) = ,

where, bi are parameter coefficients and , and

Recommendation:

From this fitted model we can suggest that the Members would be negatively affected by the dues as, if one unit of due is increased, then it would cause a decrease in Members as 27.85 units. So the policy maker has to increase the Population as well as Equipment types.