Question

use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. assume that the population has a normal distribution.
a laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s = 17.6 milligrams. construct a 95% confidence interval for the true mean cholesterol content of all such eggs.

a. 175.9 mg < μ < 194.1 mg
b. 173.2 mg < μ < 196.6 mg
c. 173.5 mg < μ < 196.5 mg
d. 173.8 mg < μ < 196.2 mg

Explanation:

Step1: Find the critical value \(t_{\alpha/2}\)

The sample size \(n = 12\), so the degrees of freedom \(df=n - 1=12-1 = 11\).
For a \(95\%\) confidence interval, \(\alpha=1 - 0.95=0.05\), and \(\alpha/2=0.025\).
From the \(t\) - distribution table, \(t_{0.025,11}=2.201\).

Step2: Calculate the margin of error \(E\)

The formula for the margin of error when the population standard deviation \(\sigma\) is unknown is \(E = t_{\alpha/2}\frac{s}{\sqrt{n}}\).
Given \(s = 17.6\), \(n = 12\), and \(t_{\alpha/2}=2.201\).

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Step3: Construct the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\).
Given \(\bar{x}=185\).

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Answer:

D. \(173.8\space mg<\mu<196.2\space mg\)