April 25, 2009 - Class Notes

A. Introduction to Hypothesis Testing

A hypothesis test is a process that uses sample statistics to test a claim about the value of a
population parameter.

A statistical hypothesis is a verbal statement, or claim, about a population parameter.
A null hypothesis H0 is a claim that contains a statement of equality such as ≤, =, or ≥.
An alternative hypothesis Ha is the complement of the null hypothesis. It is a statement that
must be true if H0 is false and the statement contains a statement of inequality such as >, ≠, or <.

A type I error occurs if the null hypothesis is rejected when it is true. For example, the null
hypothesis H0 claims that the new allergy drug lasts for 36 hours. The decision made from a
hypothesis testing is to reject H0, although H0 is true.

A type II error occurs if the null hypothesis is not rejected when it is false. In the given example,
the pharmaceutical company claims that the new allergy drug lasts for 36 hours. If the hypothesis
testing failed to reject the claim, although the actual truth is that the new drug does not last for 36
hours, we are making a type II error.
If the alternative hypothesis contains a less-than inequality symbol (<), the hypothesis test is a
left-tailed test. It can be mathematically expressed as:
H0: μ ≥ k
Ha: μ < k
If the alternative hypothesis contains a greater-than symbol (>), the hypothesis test is a righttailed
test. It can be mathematically expressed as:
H0: μ ≤ k
Ha: μ > k

If the alternative hypothesis contains the not-equal-to symbol (≠), the hypothesis test is a twotailed
test. In a two-tailed test, each tail has an area of one-half P. The hypotheses can be
mathematically expressed as:
H0: μ = k
Ha: μ ≠ k
B. Hypothesis Testing for the Mean (Large Samples)
The P-value of a hypothesis test is the probability of obtaining the sample statistic with a value as
extreme—or one that is more extreme—than the value obtained from the sample data. We reject
the null hypothesis if the P-value is less than the level of significance.
A rejection region, or critical region, of the sampling distribution is the range of values for
which the null hypothesis is not probable. If a test statistic falls in this region, the null hypothesis
is rejected.
A critical value z0 separates the rejection region from the no-rejection region.
C. Hypothesis Testing for the Mean (Small Samples)
When a sample size n is less than 30 and the random variable x is normally distributed, x follows
a t-distribution with n – 1 degrees of freedom.