Section 7.2 Hypothesis Testing for the Mean (Large Samples) Larson/Farber 4th ed.
Jan 12, 2016
Section 7.2
Hypothesis Testing for the Mean (Large Samples)
Larson/Farber 4th ed.
Section 7.2 Objectives
• Find P-values and use them to test a mean μ• Use P-values for a z-test• Find critical values and rejection regions in a normal
distribution• Use rejection regions for a z-test
Larson/Farber 4th ed.
Using P-values to Make a Decision
Decision Rule Based on P-value• To use a P-value to make a conclusion in a
hypothesis test, compare the P-value with α. If P ≤ α, then reject H0.
1. If P > α, then fail to reject H0.
Larson/Farber 4th ed.
Example: Interpreting a P-value
The P-value for a hypothesis test is P = 0.0132. What is your decision if the level of significance is
Solution:Because 0.0132 < 0.05, you should reject the null hypothesis.
Solution:Because 0.0132 > 0.01, you should fail to reject the null hypothesis.
Larson/Farber 4th ed.
2.0.01?
1.0.05?
Finding the P-value
After determining the hypothesis test’s standardized test statistic and the test statistic’s corresponding area, do one of the following to find the P-value.
a.For a left-tailed test, P = (Area in left tail).
b.For a right-tailed test, P = (Area in right tail).
c.For a two-tailed test, P = 2(Area in tail of test statistic).
Larson/Farber 4th ed.
Example: Finding the P-value
Find the P-value for a left-tailed hypothesis test with a test statistic of z = -1.99. Decide whether to reject H0 if the level of significance is α = 0.01.
z0-1.99
P = 0.0233
Solution:For a left-tailed test, P = (Area in left tail)
Because 0.0233 > 0.01, you should fail to reject H0
Larson/Farber 4th ed.
z0 1.82
Example: Finding the P-value
Find the P-value for a two-tailed hypothesis test with a test statistic of z = 1.82. Decide whether to reject H0 if the level of significance is α = 0.05.
Solution:For a two-tailed test, P = 2(Area in tail of test statistic)
Because 0.0688 > 0.05, you should fail to reject H0
0.9838
1 – 0.9656 = 0.0344
P = 2(0.0344) = 0.0688
Larson/Farber 4th ed.
Z-Test for a Mean μ
• Can be used when the population is normal and σ is known, or for any population when the sample size n is at least 30.
• The test statistic is the sample mean • The standardized test statistic is z
• When n ≥ 30, the sample standard deviation s can be substituted for σ.
Larson/Farber 4th ed.
Using P-values for a z-Test for Mean μ
1. State the claim mathematically and verbally. Identify the null and alternative hypotheses.
2. Specify the level of significance.
3. Determine the standardized test statistic.
4. Find the area that corresponds to z.
State H0 and Ha.
Identify α.
Use Table 4 in Appendix B.
Larson/Farber 4th ed.
In Words In Symbols
Using P-values for a z-Test for Mean μ
Reject H0 if P-value is less than or equal to α. Otherwise, fail to reject H0.
5. Find the P-value.a. For a left-tailed test, P = (Area in left tail).b. For a right-tailed test, P = (Area in right tail).c. For a two-tailed test, P = 2(Area in tail of test
statistic).6. Make a decision to reject or
fail to reject the null hypothesis.
7. Interpret the decision in the context of the original claim.
Larson/Farber 4th ed.
In Words In Symbols
Example: Hypothesis Testing Using P-values
A manufacturer of sprinkler systems designed for fire protection claims that the average activating temperature is at least 135˚F. To test the claim, you randomly select a sample of 32 systems and find the mean activation temperature to be 133˚F with a standard deviation of 3.3˚F. Is there enough evidence to support the claim at α = 0.01? Use a P-value.
Larson/Farber 4th ed.
Solution: Hypothesis Testing Using P-values
• H0:
• Ha:
• α = • Test Statistic:
μ ≥ 135 (Claim)
μ < 135
0.01
• Decision:
At the 10% level of significance, you have sufficient evidence to reject the manufacturer’s claim that the average activating temperature is at least 135 degrees.
0-3.43z
0.003
• P-value
0.0003 < 0.01Reject H0
Larson/Farber 4th ed.
Example: Hypothesis Testing Using P-values
An Alabama politician claims that the mean annual salary for engineering managers in Alabama is more than the national mean of $100,800. We take a random sample of 34 engineering managers salaries in Alabama. At α = 0.03, is there enough evidence to support the politician’s claim?
Larson/Farber 4th ed.
Solution: Hypothesis Testing Using P-values
• H0:
• Ha:
• α = • Test Statistic:
μ <= $100,800
μ > $100,800, (Claim)
0.03 (0.9700), z = 1.88
• Decision:
At the 3% level of significance, there is not enough evidence to support the politician’s claim!
• P-valueP ≈ 1
Larson/Farber 4th ed.
=> ~ 0
Fail to reject H0
z0 1.88
0.03
-6.58
1 > 0.03
Rejection Regions and Critical Values
Rejection region (or critical region) • 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 nonrejection region.
Larson/Farber 4th ed.
Rejection Regions and Critical Values
Finding Critical Values in a Normal Distribution1. Specify the level of significance α.2. Decide whether the test is left-, right-, or two-tailed.
3. Find the critical value(s) z0. If the hypothesis test isa. left-tailed, find the z-score that corresponds to an area of α,b. right-tailed, find the z-score that corresponds to an area of 1
– α,c. two-tailed, find the z-score that corresponds to ½α and 1 –
½α.d. Sketch the standard normal distribution. Draw a vertical
line at each critical value and shade the rejection region(s).
Larson/Farber 4th ed.
Example: Finding Critical Values
Find the critical value and rejection region for a two-tailed test with α = 0.05.
z0 z0z0
½α = 0.025 ½α = 0.025
1 – α = 0.95
The rejection regions are to the left of -z0 = -1.96 and to the right of z0 = 1.96.
z0 = 1.96-z0 = -1.96
Solution:
Larson/Farber 4th ed.
Decision Rule Based on Rejection Region
To use a rejection region to conduct a hypothesis test, calculate the standardized test statistic, z. If the standardized test statistic1. is in the rejection region, then reject H0.2. is not in the rejection region, then fail to reject H0.
z
0z0
Fail to reject H0.
Reject H0.
Left-Tailed Test
z < z0
z
0 z0
Reject Ho.
Fail to reject Ho.
z > z0
Right-Tailed Test
z0−z0
Two-Tailed Testz0z < -z0 z > z0
Reject H0
Fail to reject H0
Reject H0
Larson/Farber 4th ed.
Using Rejection Regions for a z-Test for a Mean μ
1. State the claim mathematically and verbally. Identify the null and alternative hypotheses.
2. Specify the level of significance.
3. Sketch the sampling distribution.
4. Determine the critical value(s).
5. Determine the rejection region(s).
State H0 and Ha.
Identify α.
Use Table 4 in Appendix B.
Larson/Farber 4th ed.
In Words In Symbols
Using Rejection Regions for a z-Test for a Mean μ
6. Find the standardized test statistic.
7. Make a decision to reject or fail to reject the null hypothesis.
7. Interpret the decision in the context of the original claim.
If z is in the rejection region, reject H0. Otherwise, fail to reject H0.
Larson/Farber 4th ed.
In Words In Symbols
Example: Testing with Rejection Regions
Employees in a large accounting firm claim that the mean salary of the firm’s accountants is less than that of its competitor’s, which is $45,000. A random sample of 30 of the firm’s accountants has a mean salary of $43,500 with a standard deviation of $5200. At α = 0.05, test the employees’ claim.
Larson/Farber 4th ed.
Solution: Testing with Rejection Regions
• H0:
• Ha:
• α = • Rejection Region:
μ ≥ $45,000
μ < $45,000
0.05
• Decision:At the 5% level of significance, there is not sufficient evidence to support the employees’ claim that the mean salary is less than $45,000.
• Test Statistic
z0-1.645
0.05
-1.58
-1.645
Fail to reject H0
Larson/Farber 4th ed.
Example: Testing with Rejection Regions
The U.S. Department of Agriculture reports that the mean cost of raising a child from birth to age 2 in a rural area is $10,460. You believe this value is incorrect, so you select a random sample of 900 children (age 2) and find that the mean cost is $10,345 with a standard deviation of $1540. At α = 0.05, is there enough evidence to conclude that the mean cost is different from $10,460? (Adapted from U.S. Department of Agriculture Center for Nutrition Policy and Promotion)
Larson/Farber 4th ed.
Solution: Testing with Rejection Regions
• H0:
• Ha:
• α = • Rejection Region:
μ = $10,460
μ ≠ $10,460
0.05
• Decision:At the 5% level of significance, you have enough evidence to conclude the mean cost of raising a child from birth to age 2 in a rural area is significantly different from $10,460.
• Test Statistic
z0-1.96
0.025
1.96
0.025
-1.96 1.96
-2.24
Reject H0
Larson/Farber 4th ed.
Section 7.2 Summary
• Found P-values and used them to test a mean μ• Used P-values for a z-test• Found critical values and rejection regions in a
normal distribution• Used rejection regions for a z-test• HW: 1 - 45 EO
Larson/Farber 4th ed.