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 systolic blood pressure is  = 126.07. We model their 72 blood pressures as a random sample from an underlying population with mean µ (systolic blood pressures of similar executives) .

Published national health statistics report that in the general population for males aged 35-44, systolic 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

Active ingredient in medicine

Pharmaceutical companies routinely test their products to ensure that the amount of active ingredient is within tight limits. However the chemical analysis is not precise and repeated measurements of this same specimen differ slightly. One type of analysis has errors that are normally distributed with mean 0 and standard deviation 0.0068 grams per litre.

A product is tested three times with the following concentrations of the active ingredient:

0.8403, 0.8363 and 0.8447 grams per litre

Are the data consistent with the target concentration of 0.86 grams per litre? This can be expressed as a hypothesis test comparing...

H0 :   μ  =  0.86
HA :   μ  ≠  0.86

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