students who major in business it has been re...
students who major in business it has been reported that 20.4% of incoming freshmen indicate that they will major in business or a related field. a random sample of 490 incoming college freshmen were asked their preference, and 98 replied that they were considering business as a major. part: 0 / 2 part 1 of 2 estimate the true proportion of freshman business majors with 94% confidence. round intermediate calculations to three decimal places and final answers to one decimal place. %<p<
Answer
# Explanation:
## Step1: Calculate sample proportion
The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 98$ (number of successes) and $n=490$ (sample - size). So, $\hat{p}=\frac{98}{490}=0.2$.
## Step2: Find the critical - value
For a 94% confidence interval, the significance level $\alpha = 1 - 0.94=0.06$. Then $\alpha/2=0.03$. The $z$ - value $z_{\alpha/2}=z_{0.03}$. Looking up in the standard normal table, $z_{0.03}\approx1.881$.
## Step3: Calculate the margin of error
The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.2$, $n = 490$, and $z_{\alpha/2}=1.881$ into the formula.
First, calculate $\hat{p}(1 - \hat{p})=0.2\times(1 - 0.2)=0.2\times0.8 = 0.16$.
Then, $\frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.16}{490}\approx0.000327$.
$\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\approx\sqrt{0.000327}\approx0.018$.
$E = 1.881\times0.018\approx0.034$.
## Step4: Calculate the confidence interval
The confidence interval for the population proportion $p$ is given by $\hat{p}-E<p<\hat{p} + E$.
Substitute $\hat{p}=0.2$ and $E = 0.034$ into the formula: $0.2-0.034<p<0.2 + 0.034$, which is $0.166<p<0.234$.
To convert to percentages, multiply by 100: $16.6\%<p<23.4\%$.
# Answer:
$16.6\%<p<23.4\%$