Calculate the sample mean 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
51 - 58 6
59 - 66 3
67 - 74 11
75 - 82 13
83 - 90 4
Sample Mean

Sample Variance

The sample mean is 73.8

The sample variance is 689.1

Explanation:

This a grouped data question.

A. To calculate Sample Mean, we follow the following steps:

Step A1. Calculate the midpoint of the Heart Rates in Beats per Minute

For 51 - 58, the midpoint is (54 + 55) ÷ 2 = 54.5

For 59 - 66, the midpoint is (62 + 63) ÷ 2 = 62.5

For 67 - 74 , the midpoint is (70 + 71) ÷ 2 = 70.5

For 75 - 82,  the midpoint is (78 + 79) ÷ 2 = 78.5

For 83 - 90, the midpoint is (86 + 87) ÷ 2 = 86.5

Step A2. Multiply the midpoint of the Heart Rates in Beats per Minute by the frequency

For 51 - 58, (54.5) × (6) = 327

For 59 - 66, (62.5) × (3) = 188

For 67 - 74 , (70.5) × (11) = 776

For 75 - 82,  (78.5) × (13) = 1,021

For 83 - 90, (86.5) × (4) = 346

Step A3. Add answers in Step A2 and divide by the sum of the frequency minus 1

  U  = (327 + 188 + 776  + 1,021  + 346) ÷ ((6 +  3 + 11 + 13 + 4 + 37) - 1)

    = 2,657  ÷  36  

    = 73.8

Where U represents the sample mean which is 71.8.

B. To calculate Sample Variance, we follow the following steps:

B.1. Subtract mean from each of the midpoints

For 51 - 58, (54.5) - (73.8) = -19.3

For 59 - 66, (62.5) - (73.8) = -11.3

For 67 - 74 , (70.5) - (73.8) = -144.3

For 75 - 82,  (78.5) - (73.8) = 5.7

For 83 - 90, (86.5) - (73.8) = 13.7

B.2. Add the answers above and squared  

 D^2 = (- 19  - 11 - 144 + 5 + 13)^2

        = (-157.5)^2

        = 24,806.3  

B.2. Divide D^2 by sum of the frequency minus 1

S^2 = 24,806.3 ÷ 36

  =  689.1  

Where S^2 represents the sample variance which is 689.1.

Therefore, the sample mean is 73.8 and the sample variance is 689.1

I wish you the best.

lives by a moral and ethical code