Question

A sports magazine reports that the people who watch Monday night football games on television are evenly divided between men and women. Out of a random sample of 400 people who regularly watch the Monday night game, 220 are men. Using a \(.10\) level of significance, can be conclude that the report is false?

Short answer

Based on a hypothesis test at a \(.10\) level of significance, we reject the null hypothesis as the p-value (0.0456) is less than the level of significance. Therefore, we conclude that the report claiming people who watch Monday night football games on television are evenly divided between men and women is false.

Step-by-step solution

State the null and alternative hypotheses

We want to test if the proportion of men who watch Monday night football is equal to the proportion of women. The null hypothesis (H₀) states that the proportion of men is equal to the proportion of women, meaning the proportion of men is 0.5. H₀: p = 0.5 The alternative hypothesis (H₁) states that the proportion of men is not equal to the proportion of women. H₁: p ≠ 0.5

Calculate the test statistic

We will use the sample proportion to calculate the test statistic. Test statistic (Z) = \(\frac{(p_{sample} - p_{null})}{\sqrt{\frac{p_{null}(1-p_{null})}{n}}}\) In our case: p_sample = 220/400 = 0.55 p_null = 0.5 n = 400 (sample size) Z = \(\frac{(0.55 - 0.5)}{\sqrt{\frac{0.5(1-0.5)}{400}}}\)

Calculate the Z-value

Z = \(\frac{0.05}{\sqrt{\frac{0.25}{400}}}\) ≈ 2

Calculate the p-value

Since we are performing a two-tailed test, we need to find the two-tailed p-value corresponding to the calculated Z-value. We can use a Z-table or a calculator to find this value. p-value = P(Z ≥ 2) = 2 * (1 - P(Z ≤ 2)) p-value ≈ 2 * (1 - 0.9772) ≈ 0.0456

Compare the p-value to the level of significance

Our p-value (0.0456) is less than the given level of significance (\(.10\)). Since the p-value is less than the level of significance, we reject the null hypothesis. Therefore, we can conclude that the report that the people who watch Monday night football games on television are evenly divided between men and women is false.