To estimate [tex]p[/tex], the proportion of college students who have their own bathroom, a sample of [tex]n = 100[/tex] was selected. In this sample, 60 college students had their own bathroom.

Note that:
[tex]\frac{60}{100} = 0.6[/tex]

and

[tex]\sqrt{\frac{0.6(1-0.6)}{100}} = 0.049[/tex]

For a 99.7 percent confidence interval for [tex]p[/tex], what is the value of the standard error?

A. 3
B. 100
C. 0.5
D. 99.7 percent
E. 0.049

We are given that 60 out of 100 college students have their own bathroom, so the sample proportion is

[tex]$$\hat{p} = \frac{60}{100} = 0.6.$$[/tex]

The standard error for a proportion is calculated using the formula

[tex]$$\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}},$$[/tex]

where [tex]$\hat{p}$[/tex] is the sample proportion and [tex]$n$[/tex] is the sample size.

Substitute the known values:

[tex]$$\text{SE} = \sqrt{\frac{0.6 \times (1 - 0.6)}{100}} = \sqrt{\frac{0.6 \times 0.4}{100}}.$$[/tex]

Calculate the product in the numerator:

[tex]$$0.6 \times 0.4 = 0.24.$$[/tex]

So,

[tex]$$\text{SE} = \sqrt{\frac{0.24}{100}} = \sqrt{0.0024}.$$[/tex]

Taking the square root gives

[tex]$$\text{SE} \approx 0.049.$$[/tex]

Thus, the value of the standard error is approximately [tex]$0.049$[/tex], which corresponds to choice E.