To estimate the mean, variance, and standard deviation, we first need to calculate the midpoint of each class interval, which is the value at the center of the interval. Then, we'll use these midpoints to calculate the estimated mean, variance, and standard deviation.
Given information:
Amount of Electric Bill (dollars) Number of Families
0 to less than 40 5
40 to less than 80 16
80 to less than 120 13
120 to less than 160 12
160 to less than 200 4
Step 1: Calculate the midpoints for each class interval:
Midpoint of 0 to less than 40 = (0 + 40) / 2 = 20
Midpoint of 40 to less than 80 = (40 + 80) / 2 = 60
Midpoint of 80 to less than 120 = (80 + 120) / 2 = 100
Midpoint of 120 to less than 160 = (120 + 160) / 2 = 140
Midpoint of 160 to less than 200 = (160 + 200) / 2 = 180
Step 2: Calculate the estimated mean:
Estimated Mean = (Sum of (Midpoint * Frequency)) / Total Frequency
Estimated Mean = (20 * 5 + 60 * 16 + 100 * 13 + 140 * 12 + 180 * 4) / (5 + 16 + 13 + 12 + 4)
Estimated Mean = (100 + 960 + 1300 + 1680 + 720) / 50
Estimated Mean = 5760 / 50
Estimated Mean = 115.2
Step 3: Calculate the estimated variance:
Estimated Variance = (Sum of ((Midpoint - Estimated Mean)^2 * Frequency)) / Total Frequency
Estimated Variance = ((20 - 115.2)^2 * 5 + (60 - 115.2)^2 * 16 + (100 - 115.2)^2 * 13 + (140 - 115.2)^2 * 12 + (180 - 115.2)^2 * 4) / (5 + 16 + 13 + 12 + 4)
Estimated Variance = (3835.84 * 5 + 3003.84 * 16 + 234.09 * 13 + 1980.64 * 12 + 3802.56 * 4) / 50
Estimated Variance = (19179.2 + 48061.44 + 3043.17 + 23767.68 + 15210.24) / 50
Estimated Variance = 104261.73 / 50
Estimated Variance = 2085.23
Step 4: Calculate the estimated standard deviation:
Estimated Standard Deviation = √Estimated Variance
Estimated Standard Deviation = √2085.23
Estimated Standard Deviation ≈ 45.68
So, the estimated mean is $115.2, the estimated variance is $2085.23, and the estimated standard deviation is approximately $45.68.