Question

the general social survey is an annual survey given to about 1500 u.s. adults selected at random. a recent survey contained the question \how important to your life is having a fulfilling job?\ of the 239 college graduates surveyed, 100 chose the response \very important.\ of the 108 people surveyed whose highest level of education was high school or less, 28 chose the response \very important.\ based on these data, can we conclude, at the 0.10 level of significance, that there is a difference between the proportion $p_1$ of all u.s. college graduates who would answer \very important\ and the proportion $p_2$ of all u.s. adults whose highest level of education was high school or less who would answer \very important\? perform a two - tailed test. then complete the parts below. carry your intermediate computations to three or more decimal places and round your answers as specified in the parts below. (if necessary, consult a list of formulas.) (a) state the null hypothesis $h_0$ and the alternative hypothesis $h_1$. $h_0$: $h_1$: (b) determine the type of test statistic to use. select (c) find the value of the test statistic. (round to three or more decimal places.) (d) find the p - value. (round to three or more decimal places.) (e) can we conclude that there is a difference between the two populations in the proportion who would answer \very important\? yes no

Explanation:

Step1: State hypotheses

The null hypothesis $H_0$ is that the two proportions are equal, i.e., $H_0: p_1 - p_2=0$. The alternative hypothesis $H_1$ for a two - tailed test is that the two proportions are not equal, so $H_1: p_1 - p_2
eq0$.

Step2: Determine test statistic type

We use a two - sample z - test for proportions since we are comparing two population proportions.

Step3: Calculate sample proportions

For college graduates, $n_1 = 239$ and $x_1 = 100$, so $\hat{p}_1=\frac{x_1}{n_1}=\frac{100}{239}\approx0.418$. For high - school or less educated, $n_2 = 108$ and $x_2 = 28$, so $\hat{p}_2=\frac{x_2}{n_2}=\frac{28}{108}\approx0.259$. The pooled proportion $\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{100+28}{239 + 108}=\frac{128}{347}\approx0.37$

Step4: Calculate the z - test statistic

The formula for the z - test statistic for two proportions is $z=\frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$. Substituting the values:
\[

$$\begin{align*} z&=\frac{(0.418 - 0.259)-0}{\sqrt{0.37\times(1 - 0.37)\times(\frac{1}{239}+\frac{1}{108})}}\\ &=\frac{0.159}{\sqrt{0.37\times0.63\times(\frac{108 + 239}{239\times108})}}\\ &=\frac{0.159}{\sqrt{0.2331\times\frac{347}{25812}}}\\ &=\frac{0.159}{\sqrt{0.2331\times0.0135}}\\ &=\frac{0.159}{\sqrt{0.003147}}\\ &=\frac{0.159}{0.0561}\\ &\approx2.834 \end{align*}$$

\]

Step5: Calculate the p - value

For a two - tailed z - test with $z = 2.834$, the p - value is $2\times(1 - P(Z<2.834))$. Looking up in the standard normal table, $P(Z < 2.834)\approx0.9977$, so the p - value is $2\times(1 - 0.9977)=2\times0.0023 = 0.0046$.

Step6: Make a decision

Since the p - value $0.0046<\alpha = 0.10$, we reject the null hypothesis.

Answer:

(a) $H_0: p_1 - p_2 = 0$, $H_1: p_1 - p_2
eq0$
(b) Two - sample z - test for proportions
(c) $2.834$
(d) $0.005$
(e) Yes