To determine the 95% confidence interval for the mean number of ounces of ketchup per bottle in the given sample, we follow these steps:
1. Identify the given information:
- Population mean ([tex]\(\mu\)[/tex]) = 24 ounces
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 0.4 ounces
- Sample size ([tex]\(n\)[/tex]) = 49 bottles
- Confidence level = 95%
2. Calculate the standard error of the mean (SE):
The standard error is calculated using the formula:
[tex]\[
\mathrm{SE} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\mathrm{SE} = \frac{0.4}{\sqrt{49}} = \frac{0.4}{7} = 0.0571
\][/tex]
3. Determine the z-score for the 95% confidence level:
For a 95% confidence interval, the z-score (critical value) is approximately 1.96.
4. Calculate the margin of error (ME):
The margin of error is given by:
[tex]\[
\mathrm{ME} = z \times \mathrm{SE}
\][/tex]
Substituting the values:
[tex]\[
\mathrm{ME} = 1.96 \times 0.0571 \approx 0.112
\][/tex]
5. Construct the confidence interval:
The confidence interval is calculated as:
[tex]\[
\text{Confidence Interval} = \mu \pm \mathrm{ME}
\][/tex]
Substituting the values:
[tex]\[
\text{Confidence Interval} = 24 \pm 0.112
\][/tex]
This approximate margin of error corresponds to one of the provided choices:
- A. [tex]\(24 \pm 0.229\)[/tex]
- B. [tex]\(24 \pm 0.029\)[/tex]
- C. [tex]\(24 \pm 0.057\)[/tex]
- D. [tex]\(24 \pm 0.114\)[/tex]
Given our computed margin of error of approximately 0.112, the closest match among the choices is:
D. [tex]\(24 \pm 0.114\)[/tex]
Therefore, the correct answer is:
D. [tex]\(24 \pm 0.114\)[/tex]
