"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.


e. A random sample of 15 males who are 20–29 years old results in a mean upper leg length of 46 cm. Do you find this result unusual? Why?"

Verified step by step guidance

1. Identify the population parameters given: the mean upper leg length \(\mu = 43.7\) cm and the standard deviation \(\sigma = 4.2\) cm for males aged 20 to 29.
2. Recognize that the sample size is \(n = 15\) and the sample mean is \(\bar{x} = 46\) cm.
3. Since the population distribution is approximately normal, the sampling distribution of the sample mean \(\bar{X}\) is also normal with mean \(\mu\) and standard error \(SE = \frac{\sigma}{\sqrt{n}}\).
4. Calculate the standard error using the formula: \(SE = \frac{4.2}{\sqrt{15}}\)
5. Compute the z-score to determine how many standard errors the sample mean is from the population mean using: \(z = \frac{\bar{x} - \mu}{SE} = \frac{46 - 43.7}{SE}\) Then, compare this z-score to common critical values (e.g., \(\pm 2\)) to decide if the result is unusual.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean and standard deviation, which determine the center and spread of the data. Many natural measurements, like upper leg length, approximate this distribution, allowing for probability calculations and inference.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of means from all possible samples of a given size drawn from a population. It is approximately normal if the population is normal or the sample size is large, with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Z-Score and Unusual Values

A z-score measures how many standard deviations a data point or sample mean is from the population mean. Values with z-scores beyond ±2 are often considered unusual. Calculating the z-score for the sample mean helps determine if the observed mean of 46 cm is significantly different from the population mean of 43.7 cm.