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

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=(x_1+x_2)/(n_1+n_2)\\P=(287+233)/(522+520)\\P=0.4990`

Calculate the standard error.

`{\rm S.E}=\sqrt{P(1-P)\left((1)/(n_1)+(1)/(n_2)\right)} \\{\rm S.E}=\sqrt{0.4990(1-0.4990)\left((1)/(522)+(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* S.E=(287)/(522)-(233)/(520)\pm1.96*0.0310\\p_1-p_2\pm z* S.E=0.1017\pm0.0607\\p_1-p_2\pm z* 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.