Question

Goldie's Billiards, Inc., is a retailer of billiard supplies. It stands out among billiard suppliers because of the research it does to assure its products are top notch. One experiment was conducted to measure the speed a cue ball attains when it is struck by various weighted pool cues. The conjecture is that a light cue generates faster speeds while breaking the balls at the beginning of a game of pool. Goldie's research generated the data in the file titled Breakcue.xlsx
.

a.       Make a scatter plot. Does there appear to be a linear relationship? If so, does it appear to be positive or negative?

b.       Calculate the correlation coefficient, r, using the CORREL() function in Excel:

c.       Formulate the null (H0) and alternate (HA) hypotheses for testing if a negative correlation exists between the two variables, weight of the pool cue and speed attained by the cue ball (note, this is a one-sided hypothesis test, make sure to formulate the null and alternate hypotheses appropriately):

d.       Using a = 0.025:

                                                   i.      What is the value of the test statistic, t?

                                                 ii.      What is/are the critical value(s)?

                                               iii.      To validate your results, we’ll also check our p-value. What is the p-value?

                                               iv.      Based on your p-value, do you reject or fail to reject H0?

                                                 v.      State your summary statement of the conclusion in non-technical terms.

e.       Use Excel to find the regression line and give the equation for the regression line below.

f.        Use an alpha = 0.05 to test for the significance of the regression slope coefficient. Formulate the null (H0) and alternate (HA) hypotheses:

g.       What is the p-value?

h.       Based on your p-value, do you reject or fail to reject H0?

i.         State your summary statement of the conclusion in non-technical terms.

j.         What is your value for R^2, the Coefficient of Determination? Write your answer both as a decimal and a percent.

k.       What does R^2 mean?

Weight	Speed
18.0	25.93
17.6	25.64
19.0	25.47
19.0	25.47
19.9	25.47
21.2	25.40
19.0	25.38
18.9	25.38
19.1	25.38
18.9	25.38
18.5	25.36
19.8	25.31
19.0	25.29
17.9	25.24
18.8	25.22
18.9	25.21
18.9	25.20
19.1	25.19
20.1	25.08
19.2	25.08
19.4	25.01
18.8	24.97
19.9	24.94
19.5	24.92
18.6	24.92
18.5	24.90
18.8	24.85
20.0	24.81
19.3	24.81
19.4	24.79
18.9	24.76
18.4	24.76
19.2	24.76
19.1	24.74
18.9	24.72
18.9	24.63
19.2	24.60
18.4	24.51
19.8	24.45
20.4	24.08
19.8	23.88

Answer 1

a)

Negative and weak relation between X and Y variables

b)

Correlation coefficient (r) = -0.3054 (CORREL())

c)

Hypothesis:

H0: ρ = 0

Ha: ρ < 0

d)

n=41

df=n−2=41−2=39

Test:

T stat = r*SQRT((n-2)/(1-r^2)) = -0.3054*SQRT((41-2)/(1-(-0.3054^2))) = -2.0029

T critical value = -2.0227

T stat < T critical

P value:

P value = 0.0261

P value > 0.025, Do not reject H0

Conclusion:

There is not enough evidence to conclude that there is a negative correlation exists between the two variables, weight of the pool cue and speed attained by the cue ball

e)

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.305421599
R Square	0.093282353
Adjusted R Square	0.070033183
Standard Error	0.392596799
Observations	41
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.618422869	0.618422869	4.012287385	0.05215474
Residual	39	6.011157619	0.154132247
Total	40	6.629580488
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	28.51877193	1.746929179	16.32508763	5.00667E-19	24.98527414	32.05226972	24.98527414	32.05226972
Weight(X)	-0.182882206	0.091300979	-2.003069491	0.05215474	-0.367555867	0.001791456	-0.367555867	0.001791456

Regression equation

Y = 28.5188 - 0.1829*X

f)

Hypothesis:

H0: β1 = 0

Ha: β1 not = 0

Test:

t stat = -2.003

g)

P value = 0.0521

h)

P value > 0.05, Do not reject H0

i)

There is not enough evidence to conclude that slope coefficient is significant

j)

the Coefficient of Determination (R^2) = 0.0933 or 9.33%

k)

9.33% of variation in Y variable is explained by variation in X variable