Question

In Appendix B in your book there is a table of the ages of Best Actor and Best Actress Oscar award winners. Test the claim at � = 0.05 that proportion of male winners over age 40 is greater than the proportion of female winners over the age of 40.

ACTRESSES
22
37
28
63
32
26
31
27
27
28
30
26
29
24
38
25
29
41
30
35
35
33
29
38
54
24
25
46
41
28
40
39
29
27
31
38
29
25
35
60
43
35
34
34
27
37
42
41
36
32
41
33
31
74
33
50
38
61
21
41
26
80
42
29
33
35
45
49
39
34
26
25
33
35
35
28
30
29
61
32
33
45
29
62
22
44
54

ACTORS
44
41
62
52
41
34
34
52
41
37
38
34
32
40
43
56
41
39
49
57
41
38
42
52
51
35
30
39
41
44
49
35
47
31
47
37
57
42
45
42
44
62
43
42
48
49
56
38
60
30
40
42
36
76
39
53
45
36
62
43
51
32
42
54
52
37
38
32
45
60
46
40
36
47
29
43
37
38
45
50
48
60
50
39
55
44
33

1. State the proportion of Best Actors over age 40.
   
2. State the proportion of Best Actresses over age 40.
   
3. Perform the hypothesis test.
   1. Check Conditions
      State Work here
   2. Hypotheses
      State Work here
   3. Statistic
      State Work here
   4. Critical Value(s)
      State Work here
   5. Test Statistic
      State Work here
   6. P-Value
      State Work here
   7. Decision
      State Work here
   8. Conclusion
      State Work here

Answer 1

Solution:-

a) The proportion of best actors over age 40 is 0.6322

Actors over age of 40 = 55

Total number of actors = 87

The proportion of best actors over age 40 = 55/87

The proportion of best actors over age 40 is 0.6322

b) The proportion of best actress over age 40 is 0.2644

Actress over age of 40 = 23

Total number of actress = 87

The proportion of best actress over age 40 = 23/87

The proportion of best actress over age 40 is  0.2644

c)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P_Actor< P_Actress
Alternative hypothesis: P_Actor > P_Actress

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.44828

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.075403
z = (p₁ - p₂) / SE

z = 4.88

z_critical = 1.645

Rejection region is z > 1.645

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than 4.88

Thus, the P-value = less than 0.001

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that proportion of male winners over age 40 is greater than the proportion of female winners over the age of 40.