Question

question 6
1 pts
a random sample of 600 high school students are asked about their favorite school subject, and 126 of these students say that subject is math. if a 90% confidence interval is constructed based on this sample of data, what will the interval be? as you engage in calculations, try not to do any rounding until you reach the very end of your calculations, and choose the answer below that is closest to what you calculate.
0.193 to 0.227
0.183 to 0.237
0.150 to 0.270
0.177 to 0.243
0.169 to 0.251

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 126$ (number of successes) and $n=600$ (sample size).
$\hat{p}=\frac{126}{600}=0.21$

Step2: Find critical value

For a 90% confidence interval, the significance level $\alpha=1 - 0.90=0.10$, and $\alpha/2=0.05$.
The critical value $z_{\alpha/2}$ is such that $P(Z>z_{\alpha/2}) = 0.05$. From the standard normal table, $z_{\alpha/2}=1.645$

Step3: Calculate margin of error

The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
Substitute $\hat{p}=0.21$, $n = 600$, and $z_{\alpha/2}=1.645$
$E=1.645\sqrt{\frac{0.21\times(1 - 0.21)}{600}}$
First, calculate $0.21\times(1 - 0.21)=0.21\times0.79 = 0.1659$
Then $\sqrt{\frac{0.1659}{600}}=\sqrt{0.0002765}\approx0.01663$
$E=1.645\times0.01663\approx0.02746$

Step4: Calculate confidence interval

The confidence interval is $\hat{p}-ESubstitute $\hat{p}=0.21$ and $E = 0.02746$
$0.21- 0.02746=0.18254$
$0.21 + 0.02746=0.23746$

Answer:

0.183 to 0.237