The National AIDS Behavioral Surveys interviewed a sample of adults in the cities where AIDS is most common. This sample included 789 heterosexuals who reported having more than one sexual partner in the past year. We can consider this an SRS of size 789 from the population of all heterosexuals in high-risk cities who have multiple partners. These people risk infection with the AIDS virus. Yet 292 of the respondents said they never use condoms. Is this strong evidence that more than one-third of this population never use condoms? (Use á = 0.05.)
To determine whether there is strong evidence that more than one-third of the population of heterosexuals in high-risk cities who have multiple partners never use condoms, we can perform a hypothesis test.
Hypotheses:
- Null hypothesis (H₀): The proportion of heterosexuals in this population who never use condoms is equal to or less than one-third (p ≤ 1/3).
- Alternative hypothesis (H₁): The proportion of heterosexuals in this population who never use condoms is greater than one-third (p > 1/3).
Level of significance:
- α = 0.05
Test statistic:
- We can use the z-test for proportions to perform this hypothesis test. The formula for the test statistic is z = (p̂ - p₀) / √[(p₀ * (1 - p₀)) / n]
Given information:
- Sample size (n) = 789
- Number of respondents who never use condoms (p̂) = 292
- Null hypothesis value (p₀) = 1/3 = 0.3333
Calculations:
- First, we calculate the standard error: SE = √[(p₀ * (1 - p₀)) / n]
- Then, we calculate the test statistic: z = (p̂ - p₀) / SE
Let's plug in the values and calculate the test statistic.