Question

The Army finds that the head sizes (forehead circumference) of soldiers has a normal distribution with a mean of 22.7 inches and a standard deviation of 1.1 inches.

1. Approximately 68% of soldiers have head sizes between (, ) inches.
2. Approximately 95% of soldiers have head sizes between (, ) inches.
3. Approximately 99.7% of soldiers have head sizes between (, ) inches.

Answer 1

Part a)

X ~ N ( µ = 22.7 , σ = 1.1 )
P ( a < X < b ) = 0.68
Dividing the area 0.68 in two parts we get 0.68/2 = 0.34
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.34
Area above the mean is b = 0.5 + 0.34
Looking for the probability 0.16 in standard normal table to calculate critical value Z = -0.99
Looking for the probability 0.84 in standard normal table to calculate critical value Z = 0.99
Z = ( X - µ ) / σ
-0.99 = ( X - 22.7 ) / 1.1
a = 21.611
0.99 = ( X - 22.7 ) / 1.1
b = 23.789
P ( 21.6 < X < 23.8 ) = 0.68

Part b)

X ~ N ( µ = 22.7 , σ = 1.1 )
P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate critical value Z = -1.96
Looking for the probability 0.975 in standard normal table to calculate critical value Z = 1.96
Z = ( X - µ ) / σ
-1.96 = ( X - 22.7 ) / 1.1
a = 20.544
1.96 = ( X - 22.7 ) / 1.1
b = 24.856
P ( 20.5 < X < 24.9 ) = 0.95

Part c)

X ~ N ( µ = 22.7 , σ = 1.1 )
P ( a < X < b ) = 0.997
Dividing the area 0.997 in two parts we get 0.997/2 = 0.4985
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.4985
Area above the mean is b = 0.5 + 0.4985
Looking for the probability 0.0015 in standard normal table to calculate critical value Z = -2.97
Looking for the probability 0.9985 in standard normal table to calculate critical value Z = 2.97
Z = ( X - µ ) / σ
-2.97 = ( X - 22.7 ) / 1.1
a = 19.433
2.97 = ( X - 22.7 ) / 1.1
b = 25.967
P ( 19.4 < X < 26.0 ) = 0.997