Question

A study is run to estimate the mean total cholesterol level in children 2 to 6 years of age. A sample of nine participants is selected and their total cholesterol levels are measured as follows. 185 225 240 196 175 180 194 147 223 Assume the mean and standard deviation for this sample is 196.1 and 29. Note: standard deviation of sample is provided; you need standard error. Assume that total cholesterol is normally distributed. Generate a 95% CI for the true mean total cholesterol levels in children. Interpret this confidence interval.

Answer 1

To estimate the 95% confidence interval for the true mean total cholesterol levels in children, the standard error is calculated and multiplied by the t-critical value. The resulting interval is (173.8, 218.4), suggesting where the true mean likely lies.

Step-by-step explanation:

The question asks to construct a 95% confidence interval (CI) for the true mean total cholesterol levels in children aged 2 to 6 years using the given sample data. To calculate the confidence interval, we need to use the standard error (SE) of the mean, which is the standard deviation (SD) divided by the square root of the sample size (n). In this case, SE = SD / √n = 29 / √9 = 29 / 3 = 9.67. Next, we will use the t-distribution since the sample size is small (n < 30) and the population standard deviation is unknown.

Using a t-critical value (t*) for 95% CI with degree of freedom (df) = n - 1 = 8, we find t* approximately equal to 2.306 (from t-distribution tables or calculator). Now, we compute the margin of error (ME): ME = t* × SE = 2.306 × 9.67 ≈ 22.3.

The 95% CI is then: mean ± ME = 196.1 ± 22.3, which is (173.8, 218.4). Therefore, we are 95% confident that the true mean total cholesterol level in children aged 2 to 6 lies between 173.8 mg/dL and 218.4 mg/dL. This interval gives us a range of values that plausibly includes the true mean cholesterol level for the population of interest.