To determine the approximate [tex]\(95\%\)[/tex] confidence interval for the mean number of ounces of ketchup per bottle in the sample, we need to follow these steps:
1. Identify the given parameters:
- Population mean ([tex]\(\mu\)[/tex]) = 24 ounces
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 0.2 ounces
- Sample size ([tex]\(n\)[/tex]) = 49
2. Calculate the standard error (SE):
The standard error is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the values:
[tex]\[
SE = \frac{0.2}{\sqrt{49}} = \frac{0.2}{7} = 0.028571428571428574
\][/tex]
3. Find the Z-score for a 95% confidence interval:
For a 95% confidence interval, the Z-score (critical value) is approximately 1.96 (to be more precise, it's 1.959963984540054).
4. Calculate the margin of error (ME):
The margin of error is calculated using the formula:
[tex]\[
ME = Z \times SE
\][/tex]
Plugging in the values:
[tex]\[
ME = 1.959963984540054 \times 0.028571428571428574 = 0.055998970986858694
\][/tex]
5. Determine the confidence interval:
The confidence interval is calculated using the formula:
[tex]\[
\text{Confidence Interval} = \left( \mu - ME, \mu + ME \right)
\][/tex]
Plugging in the values:
[tex]\[
\text{Confidence Interval} = \left( 24 - 0.055998970986858694, 24 + 0.055998970986858694 \right)
\][/tex]
[tex]\[
\text{Confidence Interval} = \left( 23.94400102901314, 24.05599897098686 \right)
\][/tex]
6. Match our results to the provided options:
Comparing the margin of error (ME) with the provided options, we see that our calculated ME is approximately [tex]\(0.057\)[/tex]. Therefore, the correct answer from the given options is:
[tex]\(D. 24 \pm 0.057\)[/tex]
