Question

2 Section 7.4 INFERENCE FOR MEANS
1) Example: The World Health Organization (WHO) monitors many variables to
assess a population's overall health. One of these variables is birth weight. A low
birth weight is defined as 2500 grams or less.
Suppose that babies in a town had a mean birth weight of 3,500 grams with a
standard deviation of 500 grams in 2005. This year, a random sample of 25
babies has a mean weight of 3,400 grams. Obviously, this sample weighs less on
average than the population of babies in the town in 2005. A decrease in the
town's mean birth weight could indicate a decline in overall health of the town.
Are differences this large expected in random sampling from a population with a
mean birth weight of 3,500 grams? What is the probability that a random sample
of 25 babies will have a mean birth weight of 3,400 grams or less?
We assume that the variability in individual birth weights is the same this year
as it was in 2005. In general, body measurements in a large population can be
modeled by a normal curve.
Section 7.4 DISTRIBUTION OF SAMPLE MEANS
Here are the two normal models drawn
on the same scale. Which is the
population and which is the sampling
distribution? How do you know?
d) What is the z-score for a baby that weighs 3400 grams? What is the z-score
for a sample of babies with a mean birth weight of 3400 grams? Why do your
answers make sense when you look at the normal curves in (c)?
e) What is the probability that a random sample of 25 babies weighs 3,400
grams or less? (Shade the area representing the probability in the
appropriate normal curve in (c) and give your estimate.)
f) Is the difference between 3,400g and 3,500g statistically significant? Or is
this difference what we expect to see in random sampling when the
population has a mean of 3,500g? How do you know?
3

WOULD YOU BE ABLE TO HELP ME WITH f) Is the difference between 3,400g and 3,500g statistically significant? Or is
this difference what we expect to see in random sampling when the
population has a mean of 3,500g? How do you know?

Answer 1

We need to check whether the observed mean birth weight(3400 gm) of the sampled babies in the year 2005 is statistically different from the mean birth weight of the babies in the town(3500 gm). We are given that the birth weight of the babies in the town follows a normal distribution with mean 3500 gm and standard deviation 500 gm.

Therefore, we need to assess whether the difference between 3400 gm and 3500 gm are statistically significant or is this difference what we expected to see in random sampling.

We know that the population of the babies in the town follows a normal distribution with mean 3500 gm and SD 500 gm ie N(3500,500). Therefore, the sample mean will follow a t-distribution by definition.

We need to test the following hypothesis:

(=3500 gm)  versus

(=3500)

We know that = 3400 gm

Test statistic:

this follows a t-distribution with n-1 df. We shall calculate the t-statistic

=-1.0. This is the calculated t-value. We shall calculate the tabulated value of t- at 24 df by looking at the appropriate t-table or use excel in getting the value. We need to get the value for a left tailed distribution.

The area as indicated in this figure above is the p-value for this test. ie the lower tail(since the alternative hypothesis is . The p-value obtained from Excel is 0.1636. Since we usually compare this with 0.05(which is 5% level of significance) which is more than .1636>0.05), we accept the null hypothesis that the sample mean bith weight can be considered as equal to 3500 gms.