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9.1 Elements of Hypothesis Tests 9.1.1 The Null Hypothesis and Alternative Hypothesis
9.1 Elements of Hypothesis Tests
9.1.1 The Null Hypothesis and Alternative Hypothesis
Tests of hypothesis can be classified into the following three kinds of tests:
left tailed tests
right tailed tests
two-tailed tests
Left-Tailed Tests
Suppose that the manager of a fast-food restaurant wants to know whether the average waiting time is less than 4 minutes.
We translate the above as the following hypotheses:
In general if a problem asks you to determine whether the population mean (average) is less than a specific value, say , then the hypotheses are:
The first hypothesis is called the null hypothesis, and the second hypothesis is called the alternative hypothesis.
Right-Tailed Tests
Suppose that we want to know whether the average price of history books in greater than the average price of $51.46 in the year 2000.
We translate the above as the following hypotheses:
In general if a problem asks you to determine whether the population mean (average) is greater than a specific value, say , then the hypotheses are:
Two-Tailed Tests
Suppose that the manager of the paper manufacturing company wants to know whether the average paper length is different from the required length of 11 inches.
We translate the above as the following hypotheses:
In general if a problem asks you to determine whether the population mean (average) is different from a specific value, say , then the hypotheses are:
We can summarize the above into the following table:
Left-tailed
Right-tailed
Two-tailed
Key words
Less than
More than
Different from
Hypotheses
9.1.2 Rejection Region and Nonrejection Region
In a hypothesis test, the level of significance α is the tail area or areas which are considered to be significantly small or significantly large.
Left-Tailed Tests
For left-tailed tests, the level of significance α is the left tail with area α.
z
z
α
α
The z-value associated with the left tail is called the critical value, and denoted by .
z
z
α
α
The rejection region and the nonrejection region are defined as follows:
z
z
α
α
Rejection region
Rejection region
Nonrejection region
Nonrejection region
Right-Tailed Tests
For right-tailed tests, the level of significance α is the right tail with area α. The rejection region and the nonrejection region are defined as follows:
α
α
z
z
Nonrejection region
Nonrejection region
Rejection region
Rejection region
Two-Tailed Tests
For two-tailed tests, the level of significance α includes the left-tail α/2 and right tail α/2.
α/2
α/2
α/2
α/2
z
z
The z-values associated with the left tail and the right tail are called the critical values, and are denoted by and .
α/2
α/2
α/2
α/2
z
z
The rejection region and the nonrejection region are defined as follows:
α/2
α/2
α/2
α/2
z
z
Rejection region
Rejection region
Nonrejection region
Nonrejection region
Rejection region
Rejection region