Question

A test of H0: p = 0.4 versus Ha: p > 0.4 has the test statistic z = 2.52. Part A: What conclusion can you draw at the 5% significance level? At the 1% significance level? (6 points) Part B: If the alternative hypothesis is Ha: p ≠ 0.4, what conclusion can you draw at the 5% significance level? At the 1% significance level? (4 points)

Answer 1

Using the z-distribution, it is found that:

A. In both cases for the right-tailed test, the test statistic is greater than the critical values, hence the null hypothesis is rejected in both cases, that is, it can be concluded that the proportion is greater than 0.4.

B. For a 5% significance level, the test statistic is greater than the critical value, hence it can be concluded that the proportion is different of 0.4. For a 1% significance level, the test statistic is less than the critical value, hence it cannot be concluded that the proportion is different of 0.4.

Item a:

What are the hypothesis?

• The null hypothesis is:

$H_0: p = 0.4$

• The alternative hypothesis is:

$H_1: p > 0.4$

What are the critical values?

Using a z-distribution calculator, for a right-tailed test, as we are testing if the mean is greater than a value, we have that:

• The critical value for a significance level of 0.05 is of $z^{\ast} = 1.645$.

• The critical value for a significance level of 0.01 is of $z^{\ast} = 2.327$

In both cases for the right-tailed test, the test statistic is greater than the critical values, hence the null hypothesis is rejected in both cases, that is, it can be concluded that the proportion is greater than 0.4.

Item b:

What are the hypothesis?

• The null hypothesis is:

$H_0: p = 0.4$

• The alternative hypothesis is:

$H_1: p \neq 0.4$

What are the critical values?

Using a z-distribution calculator, for a two-tailed test, as we are testing if the mean is different than a value, we have that:

• The critical value for a significance level of 0.05 is of $z^{\ast} = 1.96$.

• The critical value for a significance level of 0.01 is of $z^{\ast} = 2.576$

For a 5% significance level, the test statistic is greater than the critical value, hence it can be concluded that the proportion is different of 0.4. For a 1% significance level, the test statistic is less than the critical value, hence it cannot be concluded that the proportion is different of 0.4.

You can learn more about the use of the z-distribution to test an hypothesis at https://brainly.com/question/26283867