Question

1. Kevin Stefanski wanted to see if there is a relationship between the number of penalties the Cleveland Browns players receive a year and the number of presents they receive for the holidays. He takes a random sample of nine players. The data is presented in the table below.

Number of Penalties (X)	X2	Number of Presents (Y)	Y2	XY
0	0	15	225	0
4	16	2	4	8
5	25	5	25	25
0	0	13	169	0
16	256	1	1	16
8	64	3	9	24
42	1,764	2	4	84
1	1	80	6,400	80
10	100	37	1,369	370
ΣX = 86	ΣX2 = 2,226	ΣY = 158	ΣY2 = 8,206	ΣXY = 607

1A. Conduct a hypothesis test using the following steps. Set alpha = 0.05, two-tailed.

Step 1: State the hypotheses in symbols (including both H₀and H₁).

Step 2: Set up the criteria for making the decision. That is, find the critical value.

Step 3: Summarize the data into the appropriate test-statistic. That is, compute the correlation.

Step 4: Evaluate the Null Hypothesis (Reject or Fail to Reject?).

Step 5: State your conclusion (in words).

1B. What proportion of variability of number of presents received can be explained by the variability of number of penalties?

Answer 1

X	Y	XY	X²	Y²
0	15	0	0	225
4	2	8	16	4
5	5	25	25	25
0	13	0	0	169
16	1	16	256	1
8	3	24	64	9
42	2	84	1764	4
1	80	80	1	6400
10	37	370	100	1369
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
86	158	607	2226	8206

SSxx = Ʃx² - (Ʃx)²/n = 2226 - (86)²/9 =    1404.222222

SSyy = Ʃy² - (Ʃy)²/n = 8206 - (158)²/9 =    5432.222222

SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 607 - (86)(158)/9 =    -902.7777778

1.

Null and alternative hypothesis:  

Ho: ρ = 0  

Ha: ρ ≠ 0  

2.

df = n-2 =    7

Critical value, t_c = T.INV.2T(0.05, 7) =+ 2.3646, + 2.3646

3.

Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= -902.77778/√(1404.22222*5432.22222) = -0.3269

Test statistic :    

t = r*√(n-2)/√(1-r²) = -0.3269 *√(9 - 2)/√(1 - -0.3269²) = -0.9151

4.

Fail to reject the null hypothesis.

5.

Conclusion:

There is enough evidence to conclude that there is a relationship between the number of penalties the Cleveland Browns players receive a year and the number of presents they receive for the holidays.

1B.

Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy)

= (-902.77778)²/(1404.22222*5432.22222) = 0.1068

10.68% of variability of number of presents received can be explained by the variability of number of penalties.