Upper Leg Length The upper leg length of 20- to 29-year-old males is approximately normal with a mean length of 43.7 cm and a standard deviation of 4.2 cm.
Source: “Anthropometric Reference Data for Children and Adults: U.S. Population, 1999–2002”; Volume 361, July 7, 2005.
d. What effect does increasing the sample size have on the probability? Provide an explanation for this result.

Verified step by step guidance

1. Understand that the problem involves a normally distributed variable: upper leg length with mean \(\mu = 43.7\) cm and standard deviation \(\sigma = 4.2\) cm.
2. Recall that when we take a sample of size \(n\), the sampling distribution of the sample mean has the same mean \(\mu\) but a smaller standard deviation called the standard error, calculated as \(SE = \frac{\sigma}{\sqrt{n}}\).
3. Recognize that increasing the sample size \(n\) decreases the standard error \(SE\), making the distribution of the sample mean narrower and more concentrated around the population mean.
4. Understand that a smaller standard error means the probability that the sample mean falls within a certain range around the population mean increases, because the sample mean is less variable.
5. Conclude that increasing the sample size increases the precision of the estimate of the population mean, thereby increasing the probability that the sample mean is close to the true mean.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes how the mean values of different samples from the same population are distributed. It is approximately normal if the population is normal or the sample size is large, with a mean equal to the population mean and a standard deviation (standard error) that decreases as sample size increases.

Standard Error and Its Relationship to Sample Size

The standard error measures the variability of the sample mean and is calculated as the population standard deviation divided by the square root of the sample size. Increasing the sample size reduces the standard error, making the sample mean more precise and less variable around the population mean.

Effect of Sample Size on Probability

As sample size increases, the distribution of the sample mean becomes narrower, increasing the probability that the sample mean lies close to the population mean. This means larger samples provide more reliable estimates and reduce uncertainty in probability calculations related to the sample mean.