Math 225 Course Notes

Section 7.2: Hypothesis Testing: A Single Population Mean

Key Concepts

A hypothesis about a population mean takes the form 
  H0:  = 0
An alternative hypothesis can be one-sided. 
  HA:  < 0
or 
  HA:  > 0
Otherwise, an alternative hypothesis can be two-sided 
  HA:  does not equal 0
In any case, the hypothesis is tested by considering the sample mean and its sampling distribution.

If the population standard deviation is known, is standardized by subtracting the mean under H0 and dividing by the known SE. The result can compared to the standard normal distribution with a z test.

If the population standard deviation must be estimated by the sample standard deviation s, then the standardized statistic should be compared to a t distribution with n-1 degrees of freedom with a t test, where n is the size of the sample.

Both tests depend on the central limit theorem, and the fact that the distribution of the sample mean is approximately normal. In practice, the tests will be fine for either very large samples from any population, moderately sized samples from populations that are not strongly skewed and do not contain large outliers, or small samples from approximately normal populations.

The z Test

To test a hypothesis 
  H0:  = 0
when the SE of is known, compute 
   z = ( - 0) /
If the alternative hypothesis is 
  HA:  < 0
then the p-value is the area under the standard normal curve to the left of z.

If the alternative hypothesis is 

  HA:  > 0
then the p-value is the area under the standard normal curve to the right of z.

If the alternative hypothesis is two-sided, 

  HA:  is not equal to 0
then the p-value is twice as large as the one-sided p-value, twice the area outside of z under the standard normal curve.

The t Test

To test a hypothesis 
  H0:  = 0
when the SE of is unknown, compute 
   t = ( - 0) /
If the alternative hypothesis is 
  HA:  < 0
then the p-value is the area under the t-distribution with n-1 degrees of freedom to the left of t.

If the alternative hypothesis is 

  HA:  > 0
then the p-value is the area under the under the t-distribution with n-1 degrees of freedom to the right of t.

If the alternative hypothesis is two-sided, 

  HA:  is not equal to 0
then the p-value is twice as large as the one-sided p-value, twice the area outside of t under the t-distribution with n-1 degrees of freedom.

Example

In studies from recent years, the birthweights of infants born in Boston had population standard deviations of 20.6 ounces. 40 infants were randomly sampled from recent births in this population, and the mean of their birth weights was 114.0 ounces. Is there sufficient evidence to conclude that the mean birth weight in this population is greater than 110.0 ounces?

The hypotheses are 

H0:  = 110

HA:  > 110
The sample size is large enough that it is reasonable to conclude that the shape of the sampling distribution of is approximately normal. From previous studies, it is reasonable to conclude that the standard error of this distribution is = 3.26.

Under H0, the mean of the sampling distribution is 110. After standardizing the test statistic, we find that z = 1.23.

Since the standard error was known, this value may be compared to the standard normal distribution. The p-value is the area under the curve to the right of 1.23, or .1093. In other words, if the null hypothesis were true, we would expect to see a result at least as extreme as we did about once every ten times. This is not tremendously unusual.

If the standard error is estimated from data in the sample instead of "known" from prior studies, it is often appropriate to compare the test statistic to a t distribution instead of a normal distribution. The population should not contain any strong skewness, large outliers, or contain only a small number of possible values.
14 infants were randomly sampled from recent births in Boston, and the mean and standard deviations of their birth weights were 114.0 and 18.4 ounces respectively. Examination of the data shows that it is roughly symmetrical and has no obvious outliers or marked skewness. Is there strong evidence that the population mean is below 120 ounces?

The hypotheses are 

H0:  = 120

HA:  < 120
We do not know the standard error for exactly, so we must estimate it by 18.4 / sqrt( 14 ) = 4.92. The standardized test statistic is t = (114 - 120)/4.92 = -1.22

Since we are estimating the SE, we should compare to the t distribution instead of the normal. There are 14 - 1 = 13 degrees of freedom. The p-value is the area to the left of -1.22 under a t-distribution with 13 degrees of freedom. Since 1.22 is less than 1.350, the table in the book allows us only to say that the p-value is more than 0.10.

The data is consistent with the null hypothesis, although it would need to differ strikingly to be inconsistent for such a small sample.

Last modified: Mar 25, 1996

Bret Larget, larget@mathcs.duq.edu