## Related questions with answers

Execute these steps.

a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results.

Employ the traditional method of hypothesis testing unless otherwise specified.

The percentages of adults 25 years of age and older who have completed 4 or more years of college are $23.6 \%$ for females and $27.8 \%$ for males. A random sample of women and men who were 25 years old or older was surveyed with these results. Estimate the true difference in proportions with $95 \%$ confidence, and compare your interval with the Almanac statistics.

Women | Men | |
---|---|---|

Sample size | 350 | 400 |

No. who completed 4 or more years | 100 | 115 |

Solution

VerifiedGiven:

$c=95\%$

$x_1=100$

$n_1=350$

$x_2=115$

$n_2=400$

The sample proportion is the number of successes divided by the sample size:

$\hat{p}_1=\dfrac{x_1}{n_1}=\dfrac{100}{350}\approx 0.2857$

$\hat{p}_2=\dfrac{x_2}{n_2}=\dfrac{115}{400}=0.2875$

For confidence level $1-\alpha=0.95$, determine $z_{\alpha/2}=z_{0.025}$ using table E (look up 0.025 in the table, the z-score is then the found z-score with opposite sign):

$z_{\alpha/2}=1.96$

The endpoints of the confidence interval for $p_1-p_2$ are then:

$\begin{align*} &(\hat{p}_1-\hat{p}_2)-z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \\ &=(0.2857-0.2875)-1.96\sqrt{\dfrac{0.2857(1-0.2857)}{350}+\dfrac{0.2875(1-0.2875)}{400}} \\ &\approx -0.0667 \\ \\ &(\hat{p}_1-\hat{p}_2)+z_{\alpha/2}\cdot \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \\ &=(0.2857-0.2875)+1.96\sqrt{\dfrac{0.2857(1-0.2857)}{350}+\dfrac{0.2875(1-0.2875)}{400}} \\ &\approx 0.0631 \end{align*}$

We note that the confidence interval contains the difference of the Almanac statistics: $23.6\%-27.8\%=-4.2\%=-0.042$

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