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:

The sample standard deviation is 13.22°F

Step-by-step 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 = [tex]\frac{52620.5}{744}[/tex] = 70.73

Sample standard deviation = [tex]\frac{9822.6}{744 - 1} = \frac{9822.6}{743}[/tex] = 13.22

∴ we get

The sample standard deviation is 13.22°F

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