Question

A student wants to survey the sophomore class of 200 students about whether the school should require uniforms. A

random sample of 50 sophomores is surveyed and asked whether they support the school adopting uniforms. Of
the 50 sophomores, 12 say they would favor school uniforms. Assuming the conditions for inference have been met,
what is the 90% confidence interval for the true proportion of sophomores who favor the adoption of uniforms?
0.24 +1.96
0.24(1-0.24)
200
O 0.7601.964
0.76(1 -0.76)
200
O 0.24 +1.657
0.24(1-0.24)
50
O 0.76 +1.95
0.76(1-0.76)
50

Answer 1

Answer: C

Explanation:

Given:

Sample size (n) = 50

x = 12

`\widehat{\mathbf{p}}=\frac{\mathbf{x}}{n}=(12)/(50)=0.24`

Confidence level = 90%

α = 1 − 0.90 = 0.10

α/2 = 0.05

`\text { Critical value }\left(z_(c)\right)=z_{(\alpha)/(2)}=z_(0.05)=1.6449`

(from standard normal table)

90% Confidence interval is,

`\begin{aligned}&\text { Confidence interval }=\widehat{\mathbf{p}} \pm z_(c) * \sqrt{\frac{\hat{\mathbf{p}}(1-\hat{\mathbf{p}})}{n}} \\&\text { C. I }=0.24 \pm 1.6449 * \sqrt{(0.24(1-0.24))/(50)} \\&\text { C. I }=0.24 \pm 1.65 * \sqrt{(0.24(1-0.24))/(50)}\left\end{aligned}`

Therefore, 90% confidence interval for the true proportion of sophomores who favour the adoption of uniforms is C