Question

Consider the following hypothesis test: H0: p ? .75 Ha: p < .75 A sample of 300 items was selected. Compute the p-value and state your conclusion for each of the following sample results. Use ? = .05. Round your answers to four decimal places.

a. p = .68 p-value? Conclusion: p-value H0?

b. p = .72 p-value? Conclusion: p-value H0 ?

c. p = .70 p-value? Conclusion: p-value H0 ?

d. p = .77 p-value? Conclusion: p-value H0?

Answer 1

For each sample result, we compute the p-value and state the conclusion using the given significance level of α = 0.05. Sample results a, b, and c do not provide enough evidence to reject the null hypothesis, while sample result d also does not provide enough evidence to reject the null hypothesis.

Step-by-step explanation:

For each sample result, we need to compute the p-value and state our conclusion using the given significance level of α = 0.05.

1. a. For p = 0.68, the test statistic and p-value can be calculated as follows:

   Test statistic: z = (p - p0) / √((p0(1 - p0)) / n) = (0.68 - 0.75) / √((0.75(1 - 0.75)) / 300) = -2.727

   p-value: The p-value corresponds to the area under the standard normal distribution curve to the left of the test statistic. Using a standard normal distribution table or a calculator, we find the p-value to be approximately 0.0035.

   Conclusion: Since the p-value (0.0035) is less than the significance level (α = 0.05), we reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha). There is enough evidence to suggest that the true population proportion (p) is less than 0.75.

2. b. For p = 0.72, the test statistic and p-value can be calculated in the same manner:

   Test statistic: z = (p - p0) / √((p0(1 - p0)) / n) = (0.72 - 0.75) / √((0.75(1 - 0.75)) / 300) = -1.0914

   p-value: The p-value is the area under the standard normal distribution curve to the left of the test statistic. Using a standard normal distribution table or a calculator, we find the p-value to be approximately 0.1384.

   Conclusion: Since the p-value (0.1384) is greater than the significance level (α = 0.05), we fail to reject the null hypothesis (H0). There is not enough evidence to suggest that the true population proportion (p) is less than 0.75.

3. c. For p = 0.70:

   Test statistic: z = (p - p0) / √((p0(1 - p0)) / n) = (0.70 - 0.75) / √((0.75(1 - 0.75)) / 300) = -0.9090

   p-value: The p-value is the area under the standard normal distribution curve to the left of the test statistic. Using a standard normal distribution table or a calculator, we find the p-value to be approximately 0.1842.

   Conclusion: Since the p-value (0.1842) is greater than the significance level (α = 0.05), we fail to reject the null hypothesis (H0). There is not enough evidence to suggest that the true population proportion (p) is less than 0.75.

4. d. For p = 0.77:

   Test statistic: z = (p - p0) / √((p0(1 - p0)) / n) = (0.77 - 0.75) / √((0.75(1 - 0.75)) / 300) = 0.727

   p-value: The p-value is the area under the standard normal distribution curve to the right of the test statistic. Using a standard normal distribution table or a calculator, we find the p-value to be approximately 1 - 0.7661 = 0.2339.

   Conclusion: Since the p-value (0.2339) is greater than the significance level (α = 0.05), we fail to reject the null hypothesis (H0). There is not enough evidence to suggest that the true population proportion (p) is less than 0.75.

Answer 2

The p-value must be calculated for different sample proportions to decide whether to reject the null hypothesis in hypothesis testing. Comparing the p-value to the significance level (α) will lead to the conclusion. Specific calculations are required to find the exact p-values.

Step-by-step explanation:

In hypothesis testing, determining the p-value is crucial for deciding whether to reject the null hypothesis (H0). Given H0: p ≥ .75 and Ha: p < .75, and α = .05, we will calculate the p-value for each sample proportion and compare it to α to make our conclusion:

• a. For p = .68, if the p-value is less than α, we reject H0. Calculation required to determine exact value.
• b. For p = .72, if the p-value is less than α, we reject H0. Calculation required to determine exact value.
• c. For p = .70, if the p-value is less than α, we reject H0. Calculation required to determine exact value.
• d. For p = .77, p-value is irrelevant; since p > .75, H0 cannot be rejected based on the alternative hypothesis direction.

The exact p-values require the calculation using a relevant statistical test, such as the Z-test for proportions. Once calculated, if the p-value is less than α, the conclusion is that there is sufficient evidence to reject H0. If p-value ≥ α, then we fail to reject H0.