Confidence Interval Estimation for Male Student Weight

In this detailed analysis, we will explore the confidence interval estimation for the average weight of male students based on a sample. We have a population of 150 male students and have recently sampled 29 of them. The average weight of the sample is 75 kg with a standard deviation of 15 kg. Our goal is to determine the 95% confidence interval for the average weight of the entire population. Let's delve into the process step-by-step.

Sample Characteristics and Initial Calculations

From our sample, we have the following information:

Sample mean Mean 75 kg Sample standard deviation SD 15 kg Sample size n 29 Population size N 150

First, we need to calculate the standard error of the mean (SE).

Standard Error (SE) of the mean SD / √n

Plugging in the values:

SE 15 / √29 ≈ 13.927 kg

Find the Critical Value

The critical value (t) is derived from the t-distribution table at a 95% confidence level. For a sample size of 29, the degrees of freedom (df) are calculated as:

df n - 1 29 - 1 28

From the t-distribution table, the critical value (t) for 28 degrees of freedom at 95% confidence level is approximately 2.04840914179524.

Calculate the Margin of Error

The margin of error (ME) is calculated as:

ME t × SE

Plugging in the values:

ME 2.04840914179524 × 13.927 ≈ 28.59 kg

Given the sample mean of 75 kg, the 95% confidence interval can be calculated as:

Lower Bound Mean - ME 75 - 28.59 46.41 kg

Upper Bound Mean ME 75 28.59 103.59 kg

Therefore, with 95% confidence, we can say that the average weight of the male students in the population lies between 46.41 kg and 103.59 kg.

Soft-ware Tracked Values

The calculations discussed above are consistent with soft-ware tracked values, providing a reliable estimate for the confidence interval.

Weighted Standard Error for the Whole Population

For a more precise confidence interval, especially considering the entire population, we need to calculate the weighted standard error. The hint suggests considering the squared standard error.

The weighted standard error (SEpop) is:

SEpop √[(N - n) / N] × SD √[(150 - 29) / 150] × 15 ≈ 13.77 kg

Using this SE, we can estimate the confidence interval using either the t-distribution or the normal distribution. For simplicity, we will use the normal distribution.

ME (Normal) Z × SEpop where Z is the critical value from the standard normal distribution for a 95% confidence level, which is approximately 1.96.

Plugging in the values:

ME (Normal) 1.96 × 13.77 ≈ 27.04 kg

Therefore, the 95% confidence interval is:

Lower Bound Mean - ME (Normal) 75 - 27.04 47.96 kg

Upper Bound Mean ME (Normal) 75 27.04 102.04 kg

With this, we have a more refined estimate for the 95% confidence interval of the average weight of male students in the population.