Mathematics, 02.08.2021 21:00 imanim3851

The National Center for Education Statistics would like to test the hypothesis that the proportion of Bachelor's degrees that were earned by women equals 0.60. A random sample of 140 college graduates with Bachelor degrees found that 75 were women. The National Center for Education Statistics would like to set α = 0.10. The conclusion for this hypothesis test would be that because the absolute value of the test statistic is .

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This question involves the hypothesis test for the proportion. First we build the hypothesis, then find the test statistic, and according to the test statistic, we get the following

The conclusion for this hypothesis test would be that because the absolute value of the test statistic is less than the critical value of 1.645, we do not reject the null hypothesis that the proportion of Bachelor's degrees that were earned by women equals 0.60.

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The National Center for Education Statistics would like to test the hypothesis that the proportion of Bachelor's degrees that were earned by women equals 0.60

This means that at the null hypothesis, it is tested if the proportion is of 0.6, that is:

H_0: p = 0.6

At the alternative hypothesis, we test if the proportion is different of 0.6, that is:

$H_1: p \neq 0.6$

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Decision rule:

Two-tailed test(test if the proportion is different of a value), so we exclude the top and bottom 0.1/2 = 0.05, meaning that looking at the z-table, we find a critical value of |Z_c| = 1.645 , and the decision rule is:

Accept the null hypothesis: |Z| < 1.645

Reject the null hypothesis: |Z| 1.645

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Test statistic:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

In which X is the sample mean, $\mu$ is the value tested at the null hypothesis, $\sigma$ is the standard deviation and n is the size of the sample.

0.6 is tested at the null hypothesis:

This means that $\mu = 0.6, \sigma = 0.4$

75 out of a sample of 140:

This means that:

$n = 140, \pi = \frac{75}{140} = 0.5357$

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Value of the test statistic:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

$z = \frac{0.5357 - 0.6}{\frac{\sqrt{0.6*0.4}}{\sqrt{140}}}$

z = -1.55

So |z| = 1.55

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Decision:

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