To determine the confidence interval for the true proportion of sophomores who favor school uniforms, let's follow the steps one by one.
### Step 1: Identify the Sample Proportion
First, we calculate the sample proportion. From the survey:
- Sample size ([tex]\( n \)[/tex]) = 50
- Number of sophomores who favor uniforms ([tex]\( x \)[/tex]) = 12
The sample proportion ([tex]\( \hat{p} \)[/tex]) is:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{12}{50} = 0.24 \][/tex]
### Step 2: Calculate the Standard Error
The standard error (SE) for the sample proportion is calculated using:
[tex]\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]
Plugging in the values, we get:
[tex]\[ SE = \sqrt{\frac{0.24(1 - 0.24)}{50}} = \sqrt{\frac{0.24 \times 0.76}{50}} = \sqrt{\frac{0.1824}{50}} \approx 0.0604 \][/tex]
### Step 3: Find the Z-value for the Desired Confidence Level
For a 90% confidence level, the corresponding Z-value is approximately 1.65 (specifically, 1.645).
### Step 4: Calculate the Margin of Error
The margin of error (ME) is given by:
[tex]\[ ME = Z \times SE \][/tex]
Using the Z-value and the SE calculated:
[tex]\[ ME = 1.65 \times 0.0604 \approx 0.0994 \][/tex]
### Step 5: Determine the Confidence Interval
Finally, we determine the confidence interval for the true proportion. The confidence interval is:
[tex]\[ \text{Confidence Interval} = \hat{p} \pm ME \][/tex]
So, the lower bound is:
[tex]\[ 0.24 - 0.0994 \approx 0.1406 \][/tex]
And the upper bound is:
[tex]\[ 0.24 + 0.0994 \approx 0.3394 \][/tex]
Thus, the 90% confidence interval for the true proportion of sophomores who favor the adoption of uniforms is approximately [tex]\( (0.1406, 0.3394) \)[/tex].
### Correct Answer:
Comparing this with the provided answer choices, we identify the correct formula that fits our detailed calculations:
[tex]\[ 0.24 \pm 1.65 \sqrt{\frac{0.24(1-0.24)}{50}} \][/tex]
Thus, the correct choice is:
[tex]\[ 0.24 \pm 1.65 \sqrt{\frac{0.24(1-0.24)}{50}} \][/tex]
