Question

A class of 30 fifth-graders completed standardized reading test and received the following scores. Using the empirical rule, determine if the data is close to normally distributed.

86 103 92 115 94 102 123 81 108 93

97 105 73 94 117 83 99 101 98 94

48 106 100 134 98 149 95 67 102 107

Answer 1

Solution:

For given data, we have

Xbar = 98.8

S = 18.91979

Xbar + 1*S = 98.8 + 1*18.91979 =117.7198

Xbar – 1*S = 98.8 - 1*18.91979 =79.88021

No. of observations between Xbar ± 1*S = 24

Total no. of observations = 30

Proportion of numbers between Xbar ± 1*S = 24/30 = 0.80

Empirical probability between Xbar ± 1*S = 0.68

Now, we have

Xbar + 2*S = 98.8 + 2*18.91979 =136.6396

Xbar – 2*S = 98.8 - 2*18.91979 =60.96042

No. of observations between Xbar ± 2*S = 28

Total no. of observations = 30

Proportion of numbers between Xbar ± 2*S = 28/30 = 0.933333

Empirical probability between Xbar ± 2*S = 0.95

Now, we have

Xbar + 3*S = 98.8 + 3*18.91979 =155.5594

Xbar – 3*S = 98.8 - 3*18.91979 =42.04063

No. of observations between Xbar ± 3*S = 30

Total no. of observations = 30

Proportion of numbers between Xbar ± 3*S = 30/30 = 1.00

Empirical probability between Xbar ± 3*S = 0.997

So, it is observed that data is approximately close to normally distributed.