Question

A Gallup poll to survey the top concerns of Americans was conducted. Suppose that 387 women and 359 men were independently and randomly selected, and that 241 women and 202 men chose the state of the economy as their biggest concern. Can we conclude that the proportion of women (p1), choosing the state of the economy as their biggest concern, exceeds the proportion of men (p2)?  Use a significance level of α=0.01 for the test.

Step 1 of 6: State the null and alternative hypotheses for the test.

Step 2 of 4 : Find the values of the two sample proportions, p̂ 1p^1 and p̂ 2p^2. Round your answers to three decimal places.

Step 3 of 6: Compute the weighted estimate of p, p‾ . Round your answer to three decimal places.

Step 4 of 6: Compute the value of the test statistic. Round your answer to two decimal places.

Step 5 of 6: Find the P-value for the hypothesis test. Round your answer to four decimal places.

Step 6 of 6: Make the decision to reject or fail to reject the null hypothesis.

Answer 1

1).hypothesis:-

Ho: p1 - p2 = 0

H1 : p1 - p2 > 0

where, $p_{1}$ is the proportion of women and $p_{2}$ is the proportion of men .

our claim is the alternative hypothesis.

2). given data are,

sample size of women ($n_{1}$) = 387

sample size of men ($n_{2}$) = 359

number of women who chose the state of the economy as their biggest concern ($x_{1}$) = 241

number of men who chose the state of the economy as their biggest concern($x_{2}$) = 202

the two sample proportions be:-

241 P1 = 7 387

P2 = m = 359 = 0.563 P2 = 12 - 202

3).the weighted estimate of p be:-

$\bar{p}=\frac{x_{1}+x_{2}}{n_{1}+n_{2}}=\frac{241+202}{387+359}=0.594$

[ actually this is denoted by $\hat{p}$ ]

4). the value of the test statistic be:-

$z=\frac{\hat{p_{1}}-\hat{p_{2}}}{\sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}=\frac{0.623-0.563}{\sqrt{0.594*(1-0.594)*(\frac{1}{387}+\frac{1}{359})}}=1.67$

5). p value = 0.0475

[ P(Z > 1.67) = 1 - P(Z $\leq$ 1.67)= 1 - 0.9525 [ using standard normal table ]= 0.0475]

6). decision:-

p value = 0.0475 > 0.01 (level of significance)

so, we fail to reject the null hypothesis.

conclusion:-

there is not sufficient evidence to conclude that the proportion of women choosing the state of the economy as their biggest concern, exceeds the proportion of men at 0.01 level of significance.

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