Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu$ . 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. A confidence interval of $173.8 \mathrm { mg }

Question

Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu$ . 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. A confidence interval of $173.8 \mathrm { mg } < \mu < 196.2 \mathrm { mg }$ constructed for the true mean cholesterol
Content of all such eggs. It was assumed that the population has a normal distribution. What confidence level
Does this interval represent?
A) 98%
B) 99%
C) 80%
D) 95%

Quizplus · Accepted Answer

The confidence interval constructed is a 95% confidence interval. This is because the interval was constructed using the formula:

$ \bar x \pm t_{\alpha/2,n-1} \frac{s}{\sqrt{n}} $

where $\bar x$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2,n-1}$ is the t-value from the t-distribution with $n-1$ degrees of freedom such that the area under the curve between $-t_{\alpha/2,n-1}$ and $t_{\alpha/2,n-1}$ is equal to the chosen confidence level.

In this case, the interval was constructed using $n = 12$ and $s = 6$. The sample mean was reported as $\bar x = 185$ but was not explicitly used in the calculation of the interval. Therefore, to find the confidence level, we need to look up the t-value such that the area under the curve between $-t_{\alpha/2,11}$ and $t_{\alpha/2,11}$ is equal to 0.95. This t-value is approximately 2.201.

Thus, we can set up the following equation:

$ 185 \pm 2.201 \cdot \frac{6}{\sqrt{12}} = (173.8,196.2) $

which gives us the interval that was reported. Therefore, the confidence level is 95%.

Use the Given Degree of Confidence and Sample Data to Construct