HEAD CIRCUMFERENCE
Chapter 7 – Distribution of Sample Mean
5) Past results from the National Health Survey suggest that the head circumferences of 2-month-
old boys are normally distributed and have a mean of 41.1 cm and a standard deviation of 1.5 cm.
a) Give the shape, mean and standard deviation of the distribution of sample means for samples of
size 50.
Notice: the population is all two-month-old baby boys
The variable X is their head circumference in inches
X is Normal with
41.1
and
1.5
According to the Central Limit Theorem, the distribution of sample means is normal for any
sample size because X is normally distributed. Hence, for samples of size 50, the distribution
of
x
is also normal with
41.1
1.5 .212132
50
x
xn
b) What is the probability that a sample of fifty 2-month-old babies have a head circumference of
at most 40.048 cm?
40.048 41.1
( 40.048) ( ) ( 4.96) 0.0000004
1.5
50
P x P z P z
91.5 4
With calculator: normalcdf(-10 ,40.048,41.1, ) =0.000004= 10000000
50
A head circumference of 40.048 cm is a very unusual result if the population mean is 41.1. It
is a more common result in a population with mean lower than 41.1 – then, we can
conclude .... see ***** in part (d)
c) If the population has a mean of 41.1 cm, the probability that a sample of size fifty has a mean of
40.048 or less is ___0.0000004__________
This means that for samples of size 50, about ____4_ samples in 10 million will result in a sample
mean of 40.048 or less when the population mean is 41.1
Because this event only happens___4___ out of __10 million___ times, we consider it to be usual/
unusual.
d) What may this result suggest?
********This VERY UNUSUAL result suggests that probably this sample has been selected
from a population with mean lower than 41.1.
In reality the sample of size 50 with a mean of 40.048 is the FHED sample that we have in
our calculator (2-month-old baby girls data). This result suggests that the mean head
circumference of 2-month-old baby girls is smaller than for boys.
