Question

In a recent Pew Research poll, 287 out of 522 randomly selected men in the U.S. were able to identify Egypt when it was highlighted on a map of the Middle East. When 520 randomly selected women were asked, 233 were able to do so. (a) Construct and interpret a 95% confidence interval for the difference in the proportion of U.S. men and the proportion of U.S. women who can identify Egypt on a map. (b) Explain, in the context of this problem, what is meant by "95% confidence." (c) Based on this confidence interval, would we reject the hypothesis that there is no difference in these two proportions against a two-sided alternative at the α = 0.05 level? Justify your answer.

Answer 1

Answer:

Yes. Because the interval does not contain 0, there is convincing evidence that the true proportion of men who can identify Egypt on the map is different than the true proportion of women that can identify Egypt on a map.

Step-by-step explanation:

Answer 2

The 95% confidence interval can be interpreted by calculating the standard deviation. The standard deviation is the square root of its variance.

The 95% confidence interval is $(0.0410,0.1624)$.

Given:

The randomly selected men in U.S is $x_1=287$.

The total number of men in U.S is $n_1=522$.

The randomly selected men in Egypt is $x_2=233$.

The total number of men in Egypt is $n_1=520$.

(a)

Calculate the pooled proportion.

$P=\dfrac{x_1+x_2}{n_1+n_2}\\P=\dfrac{287+233}{522+520}\\P=0.4990$

Calculate the standard error.

${\rm S.E}=\sqrt{P(1-P)\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)} \\{\rm S.E}=\sqrt{0.4990(1-0.4990)\left(\dfrac{1}{522}+\dfrac{1}{520}\right)}\\{\rm S.E}=0.0310$

Now, from the standard normal table,

$P(-1.96<Z<1.96)=0.95$

Calculate the confidence interval.

$p_1-p_2\pm z\times S.E=\dfrac{287}{522}-\dfrac{233}{520}\pm1.96\times0.0310\\p_1-p_2\pm z\times S.E=0.1017\pm0.0607\\p_1-p_2\pm z\times S.E=(0.0410,0.1624)$

Thus, the 95% confidence interval is $(0.0410,0.1624)$.

(b)

In the given question, the meaning of the given confidence interval is that there is a 95% probability that the true difference in population proportion for the difference in proportion of US men and  proportion US women who can identify Egypt on map is $(0.0410,0.1624)$.

(c)

As the whole confidence interval lies above 0, therefore with 95% confidence we can reject the hypothesis that there is no difference in the two proportions.