Question

A recent poll asked people whether they supported passing a constitutional amendment to ban burning of the national flag using a 1 to 100 scale."1" means that they do not support passing the amendment at all and "100" means they support it completely. The sample was random (from the population of US adults) and included 986 people, 400 of whom were women, and 586 were men. The mean score for women was 80.6, while the mean score for men was 77.6.
Is the difference in means test statistically significant at the 5% level? Assume the standard deviation for both men and women is 3.5.

Answer 1

Answer:

Step-by-step explanation:

Hello!

You are asked to test the difference of the population means of the opinion of  men and women over a constitutional amendment to ban burning the national flag.

Population 1 (women)

Sample n₁= 400

X[bar]₁= 80.6

Population 2 (men)

n₂= 586

X[bar]₂= 77.6

σ₁= σ₂ = σ =3.5

To test whether there is a difference between the opinion of people about the amendment regarding their gender the hypothesis is:

H₀: μ₁ - μ₂ = 0

H₁: μ₁ - μ₂ ≠ 0

α: 0.05

The statistic to use is a pooled Z:

Z= (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂)   ~N (0;1)

                σ*√(1/n₁ + 1/n₂)

The critical region for this hypothesis is two tailed:

$Z_{\alpha /2} = Z_{0.025} = -1.96$

$Z_{1-\alpha /2} = Z_{0.975} = 1.96$

Z= (X[bar]₁ - X[bar]₂) - (μ₁ - μ₂)   =        (80.6-77.6) - 0       = 13.215

                σ*√(1/n₁ + 1/n₂)                 3.5√(1/400 + 1/586)

Since the calculated Z-value is greater than the right critical value -81.96) the decision is to reject the null hypothesis. There is significant evidence to conclude that there is no difference between the population means of the opinion about the constitutional amendment to ban the burning of the national flag of women and men.

I hope it helps!