Question

If a laboratory tested 82 chicken eggs and found that the mean amount of cholesterol was 228 milligrams with η = 19.0 milligrams, construct a 95% confidence interval for the true mean cholesterol content, μ, of all such eggs.

Answer 1

The 95% confidence interval for the true mean cholesterol content of chicken eggs is calculated to be (223.888, 232.112) milligrams, indicating that there's a 95% chance the true mean falls within this range.

Step-by-step explanation:

To construct a 95% confidence interval for the true mean cholesterol content, μ, of all such eggs, we can use the formula for the confidence interval of the mean when the population standard deviation (η) is known:

CI = μ ± Z*(η/√n)

Where:

• μ is the sample mean.
• Z* is the Z-score corresponding to the desired confidence level.
• η is the population standard deviation.
• n is the sample size.

In this case:

• μ = 228 milligrams
• η = 19.0 milligrams
• n = 82
• Z* for a 95% confidence level is approximately 1.96 (from Z-tables or standard normal distribution tables).

Substituting these values into the formula gives us:

CI = 228 ± 1.96*(19.0/√82)

CI = 228 ± 1.96*(2.098)

CI = 228 ± 4.112

Thus, the 95% confidence interval is (223.888, 232.112) milligrams.

This means that we are 95% confident that the true mean cholesterol content of all chicken eggs will fall between 223.888 milligrams and 232.112 milligrams.