Question

The mean of a normally distributed data set is 110, and the standard deviation is 15.

a) Use the Empirical Rule to find the probability that a randomly-selected data value is greater than 95.

b) Use the Empirical Rule to find the probability that a randomly-selected data value is greater than 125.

Answer 1

Explanation:

68% of all values are within 1 SD (110 ± 15).

95% of all values are within 2SD (110 ± 2×15 = 30)

99.7% of all values are within 3 SD (110 ± 3×15 = 45)

a)

95 is 110 - 15, so exactly 1 SD apart from the mean value.

68% or 0.68 is the probability of all values between 95 and 125.

so, 34% or 0.34 is the probability of all values between 95 and 110. and 50% or 0.5 is the probability of all values larger than 110.

so, the probability of all values larger than 95 is

0.34 + 0.5 = 0.84

b)

125 is 110 + 15, so exactly 1 SD apart from the mean value.

so, as per a) 34% or 0.34 is the probability of all values between 110 and 125. and 50% or 0.5 is the probability of all values larger than 110.

so, the probability of all values larger than 125 is

0.5 - 0.34 = 0.16