**II. Statistics in Medical Research; Hypothesis Testing**

Bering Research Laboratories is involved in a national study organized by the National Centers for Disease Control to determine whether a new birth control pill affects the blood pressures of women who are using it. A random sample of 100 women using the pill was selected, and the participants’ blood pressures were measured. The mean blood pressure was
x
¯
=
112.5
. If the blood pressures of all women are normally distributed, with mean
μ
=
110
and standard deviation
σ
=
12
, is the mean of this sample sufficiently far from the mean of the population of all women to indicate that the birth control pill affects blood pressure?

To answer this question, researchers at Bering Labs first need to know how the means of samples drawn from a normal population are distributed. As a result of the Central Limit Theorem, if all possible samples of size *n* are drawn from a normal population that has mean
μ
and standard deviation
σ
, then the distribution of the means of these samples will also be normally distributed with mean
μ
but with standard deviation given by

σ
x
¯
=
σ
n

To conclude whether women using this birth control pill have blood pressures that are significantly different from those of all women in the population, researchers must decide whether the mean *x* of their sample is so far from the population mean that it is unlikely that the sample was chosen from the population.

In this statistical test, the researchers will assume that the sample has been drawn from the population of women with mean
μ
=
110
unless the blood pressures collected in the sample are so different from those of the population that it is not reasonable to make this assumption. In statistics, we can say that the assumption is unreasonable if the probability that the sample mean,
x
¯
=
112.5
, could be drawn from this population, with mean 110, is less than 0.05. If this probability is less than 0.05, we say that the sample is drawn from a population with a mean different from 110, which strongly suggests that the women who have taken the birth control pill have, on average, blood pressures different from those of the women who are not using the pill. That is, there is statistically significant evidence that the pill affects the blood pressures of women.

Because 95% of all normally distributed data points lie within 2 standard deviations of the mean, researchers at Bering Labs need to determine whether the sample mean
x
¯
=
112.5
is more than 2 standard deviations from
μ
=
110
. To determine how many standard deviations
x
¯
=
112.5
is from
μ
=
110
, they compute the *z*-score for
x
¯
with the formula

z
x
¯
=
x
¯
−
μ
σ
/
n

Use this to determine whether there is sufficient evidence that the new birth control pill affects blood pressure.