Questions: Question 9 2 pts A laboratory randomly selected and tested 12 chicken eggs and found that the mean cholesterol was 246 mg with a standard deviation of 11.7 mg. Construct a 95% CI for the true mean cholesterol content of all such eggs. 237, 255 mg 238.6, 253.4 mg 238, 255 mg 239.9, 253.4 mg

Question 9
2 pts

A laboratory randomly selected and tested 12 chicken eggs and found that the mean cholesterol was 246 mg with a standard deviation of 11.7 mg. Construct a 95% CI for the true mean cholesterol content of all such eggs.
237, 255 mg
238.6, 253.4 mg
238, 255 mg
239.9, 253.4 mg

Transcript text: Question 9 2 pts A laboratory randomly selected and tested 12 chicken eggs and found that the mean cholesterol was 246 mg with a standard deviation of 11.7 mg . Construct a $95 \% \mathrm{Cl}$ for the true mean cholesterol content of all such eggs. $237,255 \mathrm{mg}$ $238.6,253.4 \mathrm{mg}$ 238, 255 mg 239.9, 253.4 mg

Solution

Solution Steps

Step 1: Given Data

We have the following data from the laboratory tests on chicken eggs:

Sample mean ($\bar{x}$): $246 \, \text{mg}$
Sample standard deviation ($s$): $11.7 \, \text{mg}$
Sample size ($n$): $12$
Confidence level: $95\%$

Step 2: Calculate the Margin of Error

To construct the confidence interval, we first need to calculate the margin of error using the formula:

\[ \text{Margin of Error} = t \cdot \frac{s}{\sqrt{n}} \]

Where:

$t$ is the t-score corresponding to the $95\%$ confidence level and $n-1$ degrees of freedom. For $n = 12$, $df = 11$, we find $t \approx 2.2$.
$s$ is the sample standard deviation.
$n$ is the sample size.

Substituting the values:

\[ \text{Margin of Error} = 2.2 \cdot \frac{11.7}{\sqrt{12}} \approx 7.4 \]

Step 3: Construct the Confidence Interval

The confidence interval is calculated as:

\[ \text{Confidence Interval} = \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right) \]

Substituting the values:

\[ \text{Confidence Interval} = \left( 246 - 7.4, 246 + 7.4 \right) = (238.6, 253.4) \]