Question

Would you favor spending more federal tax money on the arts? Of a random sample of n₁ = 204 women, r₁ = 56 responded yes. Another random sample of n₂ = 193 men showed that r₂ = 47 responded yes. Does this information indicate a difference (either way) between the population proportion of women and the population proportion of men who favor spending more federal tax dollars on the arts? Use α = 0.10. Solve the problem using both the traditional method and the P-value method. (Test the difference p₁ − p₂. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

test statistic
critical value	±
P-value

Conclusion

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men.Reject the null hypothesis, there is sufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men.    Reject the null hypothesis, there is insufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men.Fail to reject the null hypothesis, there is insufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men.

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.These two methods differ slightly.    The conclusions obtained by using both methods are the same.We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.

Answer 1

p1cap = X1/N1 = 56/204 = 0.2745
p1cap = X2/N2 = 47/193 = 0.2435
pcap = (X1 + X2)/(N1 + N2) = (56+47)/(204+193) = 0.2594

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Rejection Region
This is two tailed test, for α = 0.1
Critical value of z are -1.64 and 1.64.
Hence reject H0 if z < -1.64 or z > 1.64

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.2745-0.2435)/sqrt(0.2594*(1-0.2594)*(1/204 + 1/193))
z = 0.7

P-value Approach
P-value = 0.4839
As P-value >= 0.1, fail to reject null hypothesis.

test statistic    = 0.70  
critical value   ± 1.64
P-value       = 0.4839

Fail to reject the null hypothesis, there is insufficient evidence that the proportion of women favoring more tax dollars for the arts is different from the proportion of men.

The conclusions obtained by using both methods are the same.