Question

In a recent research poll, 83.4% of 500 randomly selected men in the U.S. knew the name of our current vice president. When 500 randomly selected women in the U.S. were asked, 79% knew the vice president’s name.




(a) Construct and interpret a 95% confidence interval for the true difference in the proportion of U.S. men and the proportion of U.S. women who know the name of our current vice president.













(b) Based on this confidence interval, is there convincing evidence that the true proportion of men who know the name of the current vice president is different than the true proportion of women that know the name the current vice president? Explain.

Answer 1

Answer:

a) $(0.834-0.79) - 1.96\sqrt{\frac{0.834(1-0.834)}{500}+\frac{0.79(1-0.79)}{500}}=-0.00435$

$(0.834-0.79) - 1.96\sqrt{\frac{0.834(1-0.834)}{500}+\frac{0.79(1-0.79)}{500}}=0.0924$

The 95% confidence interval for the true difference of proportions is given by (-0.00435;0.0924)

b) For this case since the confidence interval contians the value 0 we can't conclude that the true proportion of men who know the name of the current vice president is different than the true proportion of women that know the name the current vice president at 5% of significance

Step-by-step explanation:

Part a

Since our interval is at 95% of confidence, our significance level would be given by $\alpha=1-0.95=0.05$ and $\alpha/2 =0.025$. And the critical value would be given by:

$z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96$

The confidence interval for the difference of proportions is given by the following formula:  

$(\hat p_1 -\hat p_2) \pm z_{\alpha/2}\sqrt{\frac{\hat p_1 (1-\hat p_1)}{n_1} +\frac{\hat p_2 (1-\hat p_2)}{n_2}}$

Replacing the info given we got:

$(0.834-0.79) - 1.96\sqrt{\frac{0.834(1-0.834)}{500}+\frac{0.79(1-0.79)}{500}}=-0.00435$

$(0.834-0.79) - 1.96\sqrt{\frac{0.834(1-0.834)}{500}+\frac{0.79(1-0.79)}{500}}=0.0924$

The 95% confidence interval for the true difference of proportions is given by (-0.00435;0.0924)

Part b

For this case since the confidence interval contians the value 0 we can't conclude that the true proportion of men who know the name of the current vice president is different than the true proportion of women that know the name the current vice president at 5% of significance

Answer 2

The 95% confidence interval for the true difference of proportions is given by (-0.00435, 0.0924)

The confidence interval contain the value 0 conclude that the true proportion of men who know the name of the current vice president is different than the true proportion of women that know the name the current vice president at 5% of significance.

Given that,

In a recent research poll, 83.4% of 500 randomly selected men in the U.S. knew the name of our current vice president.

When 500 randomly selected women in the U.S. were asked, 79% knew the vice president’s name.

We have to determine,

Construct and interpret a 95% confidence interval for the true difference in the proportion of U.S. men and the proportion of U.S. women who know the name of our current vice president.

According to the question,

• Since our interval is at 95% of confidence, our significance level would be given by  And the critical value would be given by:

$\alpha = 1-0.95 = 0.5$ and,

$\dfrac{\alpha}{2} = 0.025$

And the critical value would be given by:

$Z_\frac{\alpha}{2} = -1.96 \\\\And \ Z__1-\frac{\alpha}{2} = -1.96$

The confidence interval for the difference of proportions is given by the following formula:  

$(p_1-p_2) \pm Z_\frac{\alpha}{2} \sqrt{\dfrac{p_1 (1-p_1)}{n_1}+ \frac{p_2(1-p_2)}{n_2}$

Substitute the values in the given formula;

$=(0.83-0.79) \pm 1.96\sqrt{\dfrac{0.83(1-0.83)}{500}+ \dfrac{0.79(1-0.79)}{500}}\\\\= 0.4 \pm 1.96 \sqrt{ \dfrac{0.30}{500} }\\\\= 0.4 \pm 1.96 \times 0.024\\\\= 0.4 \pm 0.048\\\\= (0.4+0.048, \ 0.4-0.048)\\\\= (0.435, \ -0.924)$

The 95% confidence interval for the true difference of proportions is given by (-0.00435, 0.0924)

• The confidence interval contain the value 0 conclude that the true proportion of men who know the name of the current vice president is different than the true proportion of women that know the name the current vice president at 5% of significance.

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