[DATA] Heights of Baseball PlayersData obtained from the National Center for Health Statistics show that men between the ages of 20 and 29 have a mean height of 69.3 inches, with a standard deviation of 2.9 inches. A baseball analyst wonders whether the standard deviation of heights of major-league baseball players is less than 2.9 inches. The heights (in inches) of 20 randomly selected players are shown in the table.

b. Compute the sample standard deviation.

Verified step by step guidance

1. Step 1: Calculate the sample mean (\( \bar{x} \)) of the given heights. Use the formula: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \( n = 20 \) is the number of players and \( x_i \) are the individual heights.
2. Step 2: Find the squared differences between each height and the sample mean. For each height \( x_i \), compute: \[ (x_i - \bar{x})^2 \]
3. Step 3: Sum all the squared differences obtained in Step 2. This gives: \[ \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
4. Step 4: Calculate the sample variance by dividing the sum of squared differences by \( n - 1 \) (degrees of freedom for sample variance): \[ S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
5. Step 5: Take the square root of the sample variance to find the sample standard deviation: \[ S = \sqrt{S^2} \]

Key Concepts

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of sample data. It is calculated by finding the square root of the variance, which is the average of the squared differences between each data point and the sample mean. This statistic helps understand how spread out the data points are around the mean.

Difference Between Population and Sample

A population includes all members of a group, while a sample is a subset of the population. When calculating statistics like standard deviation, the sample standard deviation uses n-1 in the denominator (Bessel's correction) to provide an unbiased estimate of the population standard deviation, accounting for the smaller sample size.

Hypothesis Testing for Variance or Standard Deviation

Hypothesis testing for standard deviation involves comparing the sample standard deviation to a known population standard deviation to determine if there is evidence that the sample's variability differs. This often uses the chi-square distribution to test if the sample variance is significantly less than, greater than, or different from the population variance.