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When dealing with sums of random variables, especially large samples, the Central Limit Theorem plays a crucial role. This theorem states that the distribution of the sum of a large number of independent, identically distributed variables will be approximately normal. To find the mean of the sums, we multiply the mean of the individual population by the sample size. For instance, in the cholesterol test example, the mean level is 180. Hence, for a sample size of 40, the mean of samples is calculated as:

• Mean of Sums = 180 (mean) × 40 (sample size) = 7200.

This gives the central point around which the sums of all samples are distributed. It is essential to understand this concept as it determines the expected behavior of the aggregated data in larger samples.

The standard deviation of the sums tells us how much the sums are expected to vary around their mean. For the sums of samples, it is calculated by

• multiplying the standard deviation of the individual values by the square root of the sample size.

In our cholesterol test scenario, the standard deviation is given as 20. Thus, the formula looks like this:

• Standard Deviation of Sums = 20 × \( \sqrt{40} \).
• Evaluating this gives approximately 126.4.

This value shows us the expected spread or dispersion of the sum of the sample around its mean. The larger this value, the broader the range of possible sums we expect.

Sample size has a profound effect on both the mean and the standard deviation of sums. Generally, as the sample size increases, the sums become more stable due to the Central Limit Theorem.

• Larger sample sizes lead to smaller standard errors.
• This means less variability and a tighter interval around the mean of sums.

In our exercise, a sample size of 40 balances providing a robust average and managing variability. The concept here is crucial for understanding why larger samples give more reliable estimates of population parameters. They tend to normalize irregularities that might occur in smaller samples, providing more precise and dependable analysis results.