A survey was conducted to determine the proportion of male and female

respondents who are in favor of a sales training program of a bottling
company. 120 out of 400 males and 130 out of 500 females are in favor of
the issue. Is there a significant difference in the proportion of males and
females who favor the issue? Use α = 0.10

To determine whether there is a significant difference in the proportion of males and females who favor the issue, we can use a hypothesis test.

Here's how to carry out the test:

Step 1: Define the null and alternative hypothesis:
- Null Hypothesis (H0): There is no significant difference in the proportions of males and females who favor the issue.
- Alternative Hypothesis (H1): There is a significant difference in the proportions of males and females who favor the issue.

Step 2: Decide on the level of significance (α) for the test. In this case, α = 0.10.

Step 3: Collect the data and compute the test statistic.
Here, we have the following data:
- For males: n1 = 400, X1 (number in favor) = 120
- For females: n2 = 500, X2 (number in favor) = 130

We need to calculate the sample proportions, p1_hat and p2_hat:
- p1_hat = X1 / n1 = 120 / 400 = 0.3
- p2_hat = X2 / n2 = 130 / 500 = 0.26

Step 4: Compute the test statistic.
We will use the Two-Proportion Z-Test to compare the proportions.
The test statistic formula is:
Z = (p1_hat - p2_hat) / sqrt((p_hat * (1 - p_hat) / n1) + (p_hat * (1 - p_hat) / n2))

Here, p_hat = (X1 + X2) / (n1 + n2) = (120 + 130) / (400 + 500) = 0.278

Plugging in the values:
Z = (0.3 - 0.26) / sqrt((0.278 * (1 - 0.278) / 400) + (0.278 * (1 - 0.278) / 500))

Step 5: Determine the critical value and calculate the p-value.
Since α = 0.10, we can find the critical value by looking up the Z-score associated with a one-tailed test at α = 0.10.
The critical value for a one-tailed test at α = 0.10 is 1.28.

We also need to calculate the p-value associated with the test statistic. The p-value is the probability of observing a test statistic as extreme as the one observed, assuming the null hypothesis is true.

To find the p-value, we can use a Z-table or a statistical software. Given Z = test statistic, we find the p-value associated with that Z-value.

Step 6: Make a decision and interpret the result.
- If the test statistic falls into the critical region (beyond the critical value) or if the p-value is less than α, we reject the null hypothesis.
- If the test statistic does not fall into the critical region or if the p-value is greater than α, we fail to reject the null hypothesis.

In this case, compare the calculated test statistic (Z) with the critical value (1.28) and compare the p-value with the level of significance (α = 0.10).
If Z > 1.28 or if the p-value < 0.10, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Now, you can calculate the test statistic and compare it to the critical value and interpret the result.