Answer :
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.
[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.