Question

The accompanying table lists the ages of acting award winners matched by the years in which the awards were won. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value of r. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Should we expect that there would be a​ correlation? Use a significance level of

alphaαequals=0.05.the award winners.

Construct a scatterplot. A scatterplot has a horizontal axis labeled Best Actress in years from 20 to 70 in increments of 5 and a vertical axis labeled Best Actor in years from 20 to 70 in increments of 5. Twelve points are plotted with approximate coordinates as follows: (27, 44); (30, 39); (28, 40); (58, 47); (31, 50); (34, 49); (46, 56); (30, 47); (61, 39); (22, 55); (43, 44); (54, 34).

Answer 1

Scatter plot:

X	Y	XY	X²	Y²
27	44	1188	729	1936
30	39	1170	900	1521
28	40	1120	784	1600
58	47	2726	3364	2209
31	50	1550	961	2500
34	49	1666	1156	2401
46	56	2576	2116	3136
30	47	1410	900	2209
61	39	2379	3721	1521
22	55	1210	484	3025
43	44	1892	1849	1936
54	34	1836	2916	1156

Ʃx =	464
Ʃy =	544
Ʃxy =	20723
Ʃx² =	19880
Ʃy² =	25150
Sample size, n =	12
x̅ = Ʃx/n = 464/12 =	38.6666667
y̅ = Ʃy/n = 544/12 =	45.3333333
SSxx = Ʃx² - (Ʃx)²/n = 19880 - (464)²/12 =	1938.66667
SSyy = Ʃy² - (Ʃy)²/n = 25150 - (544)²/12 =	488.666667
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 20723 - (464)(544)/12 =	-311.666667

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

= -311.66667/√(1938.66667*488.66667) = -0.3202

Null and alternative hypothesis:

Ho: ρ = 0

Ha: ρ ≠ 0

α = 0.05

Test statistic :  

t = r*√(n-2)/√(1-r²) = -0.3202 *√(12 - 2)/√(1 - (-0.3202)²) = -1.0689

df = n-2 = 10

p-value = T.DIST.2T(ABS(-1.0689), 10) = 0.3103

Conclusion:

p-value > α , Fail to reject the null hypothesis.

There is not sufficient evidence to conclude that there is a linear correlation between the two variables y.