Question

Often, frequency distributions are reported using unequal class widths because the frequencies of some groups would otherwise be small or very large. Consider the following​ data, which represent the daytime household temperature the thermostat is set to when someone is home for a random sample of households. Determine the class​ midpoint, if​ necessary, for each class and approximate the mean and standard deviation temperature.

Temp Frequency Class Midpoint
61-64 34 63
65-67 68 66.5
68-69 196 69
70 191 70.5
71-72 122 72
73-76 81 75
77-80 52 79
The sample standard deviation is _____degrees°F.

Answer 1

The sample standard deviation is 13.22°F

Explanation:

Given - Often, frequency distributions are reported using unequal

class widths because the frequencies of some groups would

otherwise be small or very large. Consider the following​ data,

which represent the daytime household temperature the

thermostat is set to when someone is home for a random sample

of households. Determine the class​ midpoint, if​ necessary, for

each class and approximate the mean and standard deviation

temperature.

Temp Frequency Class Midpoint

61-64 34 63

65-67 68 66.5

68-69 196 69

70 191 70.5

71-72 122 72

73-76 81 75

77-80 52 79

To find - The sample standard deviation is _____degrees°F.

Proof -

Temp Frequency(f) Midpoint(m) m×f ( m - 70.73 )²×f

61-64 34 63 2142 2031.6

65-67 68 66.5 4522 1216.7

68-69 196 69 13524 586.6

70 191 70.5 13465.5 10.1

71-72 122 72 8784 196.8

73-76 81 75 6075 2224.4

77-80 52 79 4108 3556.4

∑f = 744 ∑m×f = 52620.5 ∑ = 9822.6

So, Mean =
`(52620.5)/(744)` = 70.73

Sample standard deviation =
`(9822.6)/(744 - 1) = (9822.6)/(743)` = 13.22

∴ we get

The sample standard deviation is 13.22°F