Question

In May 2000, the Gallup Organization reported that 21% of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. You wish to conduct a hypothesis test to determine if the current percentage is less than 21%. You randomly select 450 adult Americans and find that 92% of them have a great deal of trust and confidence in the federal government handling domestic issues. Use a significance level of 0.05. State the null and alternative hypotheses, and perform the hypothesis test.

Answer 1

Null Hypothesis
`(\(H_0\)): \(p \geq 0.21\)`(Percentage of trust in the federal government ≥ 21%).

Alternative Hypothesis
`(\(H_1\)): \(p < 0.21\)` (Percentage of trust in the federal government < 21%).

Step-by-step explanation:

In hypothesis testing, the null hypothesis
`(\(H_0\))` assumes that there is no significant difference or change from the claimed value, while the alternative hypothesis
`(\(H_1\))` states the contrary, seeking to find evidence supporting a difference. In this scenario,
`\(H_0\)` suggests that the percentage of trust in the federal government's handling of domestic issues is 21% or higher, while
`\(H_1\)` asserts that the current percentage is less than 21%.

To perform the hypothesis test, we utilize the sample proportion
`(\(\hat{p}\))` obtained from the survey (92%) and compare it to the hypothesized population proportion
`(\(p_0\))` stated in the null hypothesis (21%). We calculate the test statistic using the z-score formula
`\(z = \frac{\hat{p} - p_0}{\sqrt{(p_0(1-p_0))/(n)}}\)`, where n is the sample size. Subsequently, we find the critical z-value at a significance level of 0.05 and compare it with the calculated z-score to determine if the result falls within the critical region to reject the null hypothesis.

Upon calculating the z-score and comparing it with the critical z-value, if the calculated z-score falls in the rejection region (i.e., less than the critical z-value), we reject the null hypothesis, indicating evidence that the current percentage of trust in the federal government handling domestic issues is less than 21%.