Question

The middle 51% of a normally distributed population lies between what two standard scores?

(Give your answers correct to two decimal places.) and You may need to use the appropriate table in Appendix B to answer this question.

Answer 1

Solution :

Since, normal distribution is symmetric, therefore the middle 51% of population will lie between the two standard values of same magnitude but will be opposite in sign.

Let the middle 51% of normally distributed population lies between the standard scores of -a and a.

Hence,

P(-a < Z < a) = 0.51

P(Z < a) - P(Z ≤ -a) = 0.51

Since, normal distribution is symmetric therefore, P(Z < a) = 1 - P(Z ≤ -a)

Hence,

1 - P(Z ≤ -a) - P(Z ≤ -a) = 0.51

-2.P(Z ≤ -a) = 0.51 - 1

-2.P(Z ≤ -a) = -0.49

P(Z ≤ -a) = 0.49/2

P(Z ≤ -a) = 0.245 .......................(1)

Using "qnorm" function of R we get, P(Z ≤ -0.69) = 0.245

Comparing, P(Z ≤ -0.69) = 0.245 and (1) we get,

-a = -0.69

a = 0.69

Hence, the middle 51% of a normally distributed population lies between the two standard scores of -0.69 and 0.69.

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