Question

You may need to use the appropriate appendix table or technology to answer this question.

Consider the following hypothesis test.

H0: p ≥ 0.75

Ha: p < 0.75

A sample of 270 items was selected. Compute the p-value and state your conclusion for each of the following sample results. Use

= 0.05.

(a)

p = 0.68

Find the value of the test statistic. (Round your answer to two decimal places.)

Answer 1

To test the hypothesis about the population proportion, calculate the z-value using the one-proportion z-test formula and find the p-value. Compare the p-value with the significance level to decide whether to reject the null hypothesis.

Step-by-step explanation:

To find the p-value and the value of the test statistic when testing a hypothesis about a population proportion, you can follow these steps:

1. First, set up the null hypothesis (H0) and alternative hypothesis (Ha). In this scenario, we have H0: p ≥ 0.75 and Ha: p < 0.75.
2. Next, calculate the test statistic using the formula for a one-proportion z-test. The formula is z = (p - p0) / √(p0(1 - p0) / n), where p is the sample proportion, p0 is the hypothesized population proportion, and n is the sample size.
3. After calculating the test statistic, use a standard normal distribution table or technology to find the p-value, which represents the probability of observing a value as extreme as, or more extreme than, the test statistic under the null hypothesis.
4. Finally, compare the p-value with the significance level (α). If the p-value is less than α, reject the null hypothesis; otherwise, fail to reject it.

For part (a), with p = 0.68, the sample size n = 270, and the hypothesized proportion p0 = 0.75, the test statistic is calculated as follows:

z = (0.68 - 0.75) / √(0.75(1 - 0.75) / 270)

=2.65

After getting the z-value, find the corresponding p-value and compare it with α = 0.05 to make a decision.