Tests about numerical populations

The most important characteristic of a numerical population is usually its mean, µ. Hypothesis tests therefore usually question the value of this parameter.

Blood pressure of executives

The medical director of a large company looks at the medical records of 72 male executives aged between 35 and 44 and observes that their mean blood pressure is  = 126.07. We model these 72 blood pressures as a random sample from an underlying population with mean µ (blood pressures of similar executives) .

Published national health statistics report that in the general population for males aged 35-44, blood pressures have mean 128 and standard deviation 15. Do the executives conform to this population? Focusing on the mean of the blood pressure distribution, this can be expressed as the hypotheses,

H0 :   μ  =  128
HA :   μ  ≠  128

Filling milk containers

In a bottling plant, plastic containers are filled with a nominal 2 litres of milk. However the containers are filled so quickly that it is impossible to ensure that each contains exactly 2 litres. The volume of milk in a container is approximately normally distributed with standard deviation 0.005 litres, and the machinery is adjusted to give a mean volume of 2.012 litres. (Using the normal distribution, you can check that only 1% of containers should contain less than the nominal 2 litres of milk.)

At regular intervals, twelve containers are sampled and the volume of milk in each is measured accurately to assess whether the machinery needs adjustment. (Overfilling wastes milk, but consistent underfilling is illegal.) One sample is shown below.

Volume of milk (litres)
2.024
2.015
2.022
2.025
2.008
2.024
2.021
2.018
2.020
2.023
2.005
2.016

Are the data consistent with the target mean volume of 2.012 litres? This can be expressed as a hypothesis test comparing...

H0 :   μ  =  2.012
HA :   μ  ≠  2.012

Null and alternative hypotheses

Both of the above examples involve tests of hypotheses

H0 :   μ  =  μ0
HA :   μ  ≠  μ0

where µ0 is the constant that we think may be the true mean. These are called two-tailed tests. In other situations, the alternative hypothesis may involve only high (or low) values of µ (one-tailed tests), such as

H0 :   μ  =  μ0
HA :   μ  >  μ0