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9.2 Hypothesis Tests for Population Mean (Population Standard Deviation is Known) As
9.2 Hypothesis Tests for Population Mean (Population Standard Deviation is Known)
As in section 8.1, if the population standard deviation σ is known, then we use the z-table to find the critical value.
Z-Statistic
The z-statistic is defined as
Note: This formula is the same as the one in Section 7.2.
Critical Value Approach
If the z-statistic is in the rejection region (tail area), then there is evidence the population mean is less than (or more than, or different from)……
If the z-statistic is in the nonrejection region (not tail area), then there is no evidence the population mean is less than (or more than, or different from)…… (Or we do not know……)
9.2.1 Left-Tailed Tests
Example Last year, a restaurant had a mean service time of 163.9 seconds. Suppose that the restaurant has embarked on a quality improvement effort to reduce the service time. To test whether the current service time is less than 163.9 seconds, a sample of 36 customers is selected and the sample mean is 156 seconds. Suppose the population standard deviation σ = 20 seconds.
What is the population? What is the sample?
At α = 0.05 level of significance, determine whether there is evidence that the population mean service time is less than 163.9 seconds.
Solution
The population is all the customers. The sample is the 36 customers.
This is a left-tailed test because the problem asks whether the population mean is less than 163.9.
Step 1 State the null hypothesis and the alternative hypothesis.
Step 2 Compute the z -statistic
Step 3 Find the critical value to determine the rejections of rejection and nonrejection.
Since the left-tail area α = 0.05, we can use the z-table to find the critical value -1.65. Then we have the following rejection region and nonrejection region:
-1.65
-1.65
α = 0.05
α = 0.05
z
z
z
0.05
-1.6
0.05
Step 4 Determine whether the z-statistic (in Step 2) is in the rejection region or the nonrejection region. Draw conclusion.
z = -2.37
z = -2.37
α = 0.05
α = 0.05
-1.65
-1.65
z
z
The z-statistic is in the rejection region (tail area), so there is evidence that the population mean service time µ is less than 163.9 seconds.
9.2.2 Right-Tailed Tests
Example In 2000, the mean retail price of all history books was $51.46. In 2009, a person wants to know whether the mean price has increased. A sample of 40 history books is selected and the sample mean price is $52.89.
What is the population? What is the sample?
At α = 0.1 level of significance, determine whether there is evidence that the mean retail price of all history book in 2009 is greater than the price of $51.46 in 2000. (Assume σ = $7.61 for retail price in 2000.)
Solution
The population is all the history books. The sample is the 40 books selected.
(b)
z
0.08
-1.2
0.1
α = 0.1
α = 0.1
1.28
1.28
z
z
z = 1.19
z = 1.19
The z-statistic is in the non-rejection region (not in tail area), so there is no evidence that the population mean retail price is greater than $51.46 in 2000. (Or we do not know whether the population mean retail price is greater than $51.46 in 2000.)
Note: We can use the following symmetry to find the critical value. Using the z-table, we can find that
-1.28 corresponds to the left tail 0.1. Then by symmetry 1.28 corresponds to the right tail 0.1:
0.1
0.1
0.1
0.1
Implies
Implies
1.28
1.28
-1.28
-1.28
9.2.3 Two-Tailed Tests
Example Suppose a factory manufactures papers. A worker wants to know whether the mean length of the papers produced is different from 11 inches. A sample of = 100 sheets is selected and the sample average (mean) length is = 11.003. Suppose the population standard deviation σ = 0.015 inch.
What is the population? What is the sample?
At the α = 0.05 level of significance, determine whether there is evidence that the population mean paper length is different from the expected 11 inches.
Solution
The population is all sheets of paper produced. The sample is the 100 sheets of paper selected.
(b)
Using the z-table, we can find that the critical values are -1.96 and 1.96:
α/2 =0.025
α/2 =0.025
α/2 =0.025
α/2 =0.025
-1.96 1.96
-1.96 1.96
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
z = 2
z = 2
The z-statistic is in the rejection region (tail area), so there is evidence that population mean paper length is different from 11 inches.