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9.3 Hypothesis Test for Population Mean (Population Standard Deviation is Unknown) In
9.3 Hypothesis Test for Population Mean (Population Standard Deviation is Unknown)
In many cases, we do not know the population standard deviation σ. Then we have to use the sample standard deviation s as an estimate of σ. Correspondingly, we have to use the t-table (instead of the z-table).
The hypothesis test in section 9.2 is called the z-test (because it uses the z-table). In this section, the hypothesis test is called the t-test (because it uses the t-table).
Test Statistic
The t-statistic is defined as
9.3.1 Left-Tailed Tests
Example A major car manufacturer wants to know whether the mean emission level of its engines is less than 20 parts per million of carbon. A sample of 10 engines is selected. The sample mean is 17.17 and the sample standard deviation is s = 3.98.
What is the population? What is the sample?
At α = 0.01 level of significance, is there evidence that this type of engine meets the pollution standard less than 20 parts per million of carbon?
Solution
The population is all the engines of the manufacturer. The sample is the 10 engines selected.
df = n – 1
0.01
10 – 1 = 9
2.821
α = 0.01
α = 0.01
-2.821
-2.821
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
t = -2.25
t = -2.25
The t-statistic is in the nonrejection region (not in tail area), so there is no evidence that this type of engine meets the pollution standard of less than 20 parts per million of carbon.
9.3.2 Right-Tailed Tests
Example A lake is classified as nonacidic if it has a pH level greater than 6. A researcher wants to know whether the mean pH level of the lakes of the Southern Alps is nonacidic. A sample of 15 lakes is selected. The sample mean pH level is 6.6 and the sample standard deviation is s = 0.672.
What is the population? What is the sample?
At α = 0.05 level of significance, determine whether there is evidence that the population mean pH level is greater than 6.
Solution
The population is all the lakes in the Southern Alps. The sample is the 15 lakes.
(b)
df = n – 1
0.05
15 – 1 = 14
1.761
α = 0.05
α = 0.05
1.761
1.761
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
t = 3.46
t = 3.46
The t-statistic is in the rejection region (in tail area), so there is evidence the population mean pH level is greater than 6.
9.3.3 Two-Tailed Tests
Example In 2001, the mean household expenditure for energy was $1493, according to data obtained from the U.S. Energy Information Administration. An economist wanted to know whether this amount has changed significantly from its 2001 level. In a sample of 29 households, he found the mean expenditure for energy during the most recent year to be $1618, with sample standard deviation $321.
What is the population? What is the sample?
At α = 0.05 level of significance, determine whether there is evidence that the mean household expenditure for energy has changed from its 2001 level of $1493.
Solution
The population is all the households. The sample is the 29 households.
(b)
df = n – 1
0.025
29 – 1 = 28
2.048
α/2 = 0.025
α/2 = 0.025
α/2 = 0.025
α/2 = 0.025
-2.048 2.048
-2.048 2.048
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
t = 2.10
t = 2.10
The t-statistic is in the rejection region (tail area), so there is evidence that the mean household expenditure for energy has changed from its 2001 level of $1493.