A Note on Hypothesis Testing
A NOTE ON HYPOTHESIS TESTING
|Significance Level |One-Sided Test |Two-Sided Test |
|0.10 |1.285 |1.645 |
|0.05 |1.645 |1.960 |
|0.01 |2.33 |2.575 |
Part A. Single-Sample Inference
1. Test Statistic for the Population Mean ((): Large Sample Test
H0: ( = (0 H1: ( ( (0
Test statistic:
[pic], where (0 = hypothesized value of (
Note that: Sample mean = [pic] Mean of the sample mean: E([pic]) = ( Standard error of [pic] = [pic]
Example 1
As part of a survey to determine the extent of required in-cabin storage capacity, a researcher needs to test the null hypothesis that the average weight of carry-on baggage per person is μ 0 = 12 pounds, versus the alternative hypothesis that the average weight is not 12 pounds. The analyst wants to test the null hypothesis at α = 0.05. The data collected for this study are:
n = 144; [pic]= 14.6; s = 7.8; For α = 0.05, critical values of z are ±1.96
H0: μ = 12 [two-tailed test]
H1: μ ≠ 12
[pic]
Since the test statistic falls in the upper rejection region, H0 is rejected, and we may conclude that the average amount of carry-on baggage is more than 12 pounds.
Example 2
The EPA sets limits on the concentrations of pollutants emitted by various industries. Suppose that the upper allowable limit on the emission of vinyl chloride is set at an average of 55 ppm within a range of two miles around the plant emitting this chemical. To check compliance with this rule, the EPA collects a random sample of 100 readings at different times and dates within the two-mile range around the plant. The findings are that the
|Significance Level |One-Sided Test |Two-Sided Test |
|0.10 |1.285 |1.645 |
|0.05 |1.645 |1.960 |
|0.01 |2.33 |2.575 |
Part A. Single-Sample Inference
1. Test Statistic for the Population Mean ((): Large Sample Test
H0: ( = (0 H1: ( ( (0
Test statistic:
[pic], where (0 = hypothesized value of (
Note that: Sample mean = [pic] Mean of the sample mean: E([pic]) = ( Standard error of [pic] = [pic]
Example 1
As part of a survey to determine the extent of required in-cabin storage capacity, a researcher needs to test the null hypothesis that the average weight of carry-on baggage per person is μ 0 = 12 pounds, versus the alternative hypothesis that the average weight is not 12 pounds. The analyst wants to test the null hypothesis at α = 0.05. The data collected for this study are:
n = 144; [pic]= 14.6; s = 7.8; For α = 0.05, critical values of z are ±1.96
H0: μ = 12 [two-tailed test]
H1: μ ≠ 12
[pic]
Since the test statistic falls in the upper rejection region, H0 is rejected, and we may conclude that the average amount of carry-on baggage is more than 12 pounds.
Example 2
The EPA sets limits on the concentrations of pollutants emitted by various industries. Suppose that the upper allowable limit on the emission of vinyl chloride is set at an average of 55 ppm within a range of two miles around the plant emitting this chemical. To check compliance with this rule, the EPA collects a random sample of 100 readings at different times and dates within the two-mile range around the plant. The findings are that the