Questions: Calculate the sample standard deviation and sample variance for the following frequency distribution of heart rates for a sample of American adults. If necessary, round to one more decimal place than the largest number of decimal places given in the data. Heart Rates in Beats per Minute Class Frequency 56-62 10 63-69 5 70-76 5 77-83 8 84-90 13

Solution Steps

The midpoints for each class in the frequency distribution are calculated as follows:

• For the class \(56-62\): \[ \text{Midpoint} = \frac{56 + 62}{2} = 59.0 \]
• For the class \(63-69\): \[ \text{Midpoint} = \frac{63 + 69}{2} = 66.0 \]
• For the class \(70-76\): \[ \text{Midpoint} = \frac{70 + 76}{2} = 73.0 \]
• For the class \(77-83\): \[ \text{Midpoint} = \frac{77 + 83}{2} = 80.0 \]
• For the class \(84-90\): \[ \text{Midpoint} = \frac{84 + 90}{2} = 87.0 \]

The midpoints are: \[ [59.0, 59.0, 59.0, 59.0, 59.0, 66.0, 66.0, 66.0, 66.0, 66.0, 73.0, 73.0, 73.0, 73.0, 73.0, 80.0, 80.0, 80.0, 80.0, 80.0, 80.0, 80.0, 80.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0, 87.0] \]

The mean \( \mu \) is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} \] Where:

• \( \sum_{i=1}^N x_i = 3056.0 \)
• \( N = 41 \)

Thus, \[ \mu = \frac{3056.0}{41} = 74.5 \]

The sample variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] Where:

• \( \sigma^2 = 126.2 \)

The sample standard deviation \( \sigma \) is then: \[ \sigma = \sqrt{126.2} \approx 11.2 \]

Final Answer