Question

. A normal distribution has mean = 127 and standard deviation = 21 . Give limits, symmetric ? ? about the mean, within which 95% of the population will lie: a. Individual observations. b. Means of 4 observations. c. Means of 16 observations. d. Means of 100.

Answer 1

(a)

= 127

= 21

Middle 95% corresponds to area = 0.95/2 = 0.475 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 1.96

Low Side:

Z = - 1.96 = (X - 127)/21

So,

X = 127 - (1.96 X 21) = 127 - 41.16 = 85.84

High Side:

Z = 1.96 = (X - 127)/21

So,

X = 127 + (1.96 X 21) = 127 + 41.16 = 168.16

So,

The limits are given by:

85.84 to 168.16

(b)

= 127

= 21

n = 4

SE = /

= 21/ = 10.5

Middle 95% corresponds to area = 0.95/2 = 0.475 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 1.96

Low Side:

Z = - 1.96 = ( - 127)/10.5

So,

= 127 - (1.96 X 10.5) = 127 - 20.58 = 106.42

High Side:

Z = 1.96 = ( - 127)/10.5

So,

= 127 + (1.96 X 10.5) = 127 + 20.58 = 147.58

So,

The limits are given by:

106.42 to 147.58

(c)

= 127

= 21

n = 16

SE = /

= 21/ = 5.25

Middle 95% corresponds to area = 0.95/2 = 0.475 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 1.96

Low Side:

Z = - 1.96 = ( - 127)/5.25

So,

= 127 - (1.96 X 5.25) = 127 - 10.29= 116.71

High Side:

Z = 1.96 = ( - 127)/5.25

So,

= 127 + (1.96 X 5.25) = 127 + 10.29 = 137.29

So,

The limits are given by:

116.71 to 137.29

(d)

= 127

= 21

n = 100

SE = /

= 21/ = 2.1

Middle 95% corresponds to area = 0.95/2 = 0.475 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 1.96

Low Side:

Z = - 1.96 = ( - 127)/2.1

So,

= 127 - (1.96 X 2.1) = 127 - 4.116 = 122.884

High Side:

Z = 1.96 = ( - 127)/2.1

So,

= 127 + (1.96 X 2.1) = 127 + 4.116 = 131.116

So,

The limits are given by:

122.884 to 131.116