1. 8 Chapter Hypothesis Testing with Two Samples
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2. Chapter Outline
8.1 Testing the Difference Between Means (Large Independent Samples)
8.2 Testing the Difference Between Means (Small Independent Samples)
8.3 Testing the Difference Between Means (Dependent Samples)
8.4 Testing the Difference Between Proportions
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3. Testing the Difference Between Means (Large Independent Samples)
Section 8.1
Testing the Difference Between Means (Large Independent Samples)
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4. Section 8.1 Objectives
Determine whether two samples are independent or dependent
Perform a two-sample z-test for the difference between two means μ1 and μ2 using large independent samples
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5. Two Sample Hypothesis Test
Compares two parameters from two populations.
Sampling methods:
Independent Samples
The sample selected from one population is not related to the sample selected from the second population.
Dependent Samples (paired or matched samples)
Each member of one sample corresponds to a member of the other sample.
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6. Independent and Dependent Samples
Independent Samples
Dependent Samples
Sample 1
Sample 2
Sample 1
Sample 2
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7. Example: Independent and Dependent Samples
Classify the pair of samples as independent or dependent.
Sample 1: Weights of 65 college students before their freshman year begins.
Sample 2: Weights of the same 65 college students after their freshmen year.
Solution: Dependent Samples (The samples can be paired with respect to each student)
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8. Example: Independent and Dependent Samples
Classify the pair of samples as independent or dependent.
Sample 1: Scores for 38 adults males on a psycho- logical screening test for attention-deficit hyperactivity disorder.
Sample 2: Scores for 50 adult females on a psycho- logical screening test for attention-deficit hyperactivity disorder.
Solution: Independent Samples (Not possible to form a pairing between the members of the samples; the sample sizes are different, and the data represent scores for different individuals.)
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9. Two Sample Hypothesis Test with Independent Samples
Null hypothesis H0
A statistical hypothesis that usually states there is no difference between the parameters of two populations.
Always contains the symbol ≤, =, or ≥.
Alternative hypothesis Ha
A statistical hypothesis that is true when H0 is false.
Always contains the symbol >, ≠, or <.
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10. Two Sample Hypothesis Test with Independent Samples
Ha: μ1 ≠ μ2
H0: μ1 ≤ μ2
Ha: μ1 > μ2
H0: μ1 ≥ μ2
Ha: μ1 < μ2
Regardless of which hypotheses you use, you always assume there is no difference between the population means, or μ1 = μ2.
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11. Two Sample z-Test for the Difference Between Means
Three conditions are necessary to perform a z-test for the difference between two population means μ1 and μ2.
The samples must be randomly selected.
The samples must be independent.
Each sample size must be at least 30, or, if not, each population must have a normal distribution with a known standard deviation.
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12. Two Sample z-Test for the Difference Between Means
If these requirements are met, the sampling distribution for (the difference of the sample means) is a normal distribution with
Mean:
Standard error:
Sampling distribution for :
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13. Two Sample z-Test for the Difference Between Means
Test statistic is
The standardized test statistic is
When the samples are large, you can use s1 and s2 in place of σ1 and σ2. If the samples are not large, you can still use a two-sample z-test, provided the populations are normally distributed and the population standard deviations are known.
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14. Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples)
In Words In Symbols
State the claim mathematically and verbally. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the critical value(s).
Determine the rejection region(s).
State H0 and Ha.
Identify α.
Use Table 4 in Appendix B.
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15. Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples)
In Words In Symbols
Find the standardized test statistic and sketch the sampling distribution.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If z is in the rejection region, reject H0. Otherwise, fail to reject H0.
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16. Example: Two-Sample z-Test for the Difference Between Means
A credit card watchdog group claims that there is a difference in the mean credit card debts of households in New York and Texas. The results of a random survey of 250 households from each state are shown below. The two samples are independent. Do the results support the group’s claim? Use α = (Adapted from PlasticRewards.com)
New York
Texas
s1 = $
s2 = $
n1 = 250
n2 = 250
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17. Solution: Two-Sample z-Test for the Difference Between Means
Ha:
α =
n1= , n2 =
Rejection Region:
μ1 = μ2
μ1 ≠ μ2 (claim)
Test Statistic:
0.05
250
250
Decision:
Fail to Reject H0.
At the 5% level of significance, there is not enough evidence to support the group’s claim that there is a difference in the mean credit card debts of households in New York and Texas.
z = –1.11
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18. Example: Using Technology to Perform a Two-Sample z-Test
A travel agency claims that the average daily cost of meals and lodging for vacationing in Texas is less than the same average costs for vacationing in Virginia. The table shows the results of a random survey of vacationers in each state. The two samples are independent. At α = 0.01, is there enough evidence to support the claim? (Adapted from American Automobile Association)
Texas (1)
Virginia (2)
s1 = $18
s2 = $24
n1 = 50
n2 = 35
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19. Solution: Using Technology to Perform a Two-Sample z-Test
TI-83/84 setup:
H0:
Ha:
μ1 ≥ μ2
μ1 < μ2
(claim)
Calculate:
Draw:
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20. Solution: Using Technology to Perform a Two-Sample z-Test
Rejection Region:
Decision:
Fail to Reject H0 .
At the 1% level of significance, there is not enough evidence to support the travel agency’s claim. z
0.01
–2.33
–1.25
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21. Section 8.1 Summary
Determined whether two samples are independent or dependent
Performed a two-sample z-test for the difference between two means μ1 and μ2 using large independent samples
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22. Testing the Difference Between Means (Small Independent Samples)
Section 8.2
Testing the Difference Between Means (Small Independent Samples)
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23. Section 8.2 Objectives
Perform a t-test for the difference between two population means μ1 and μ2 using small independent samples
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24. Two Sample t-Test for the Difference Between Means
If samples of size less than 30 are taken from normally-distributed populations, a t-test may be used to test the difference between the population means μ1 and μ2.
Three conditions are necessary to use a t-test for small independent samples.
The samples must be randomly selected.
The samples must be independent.
Each population must have a normal distribution.
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25. Two Sample t-Test for the Difference Between Means
The test statistic is
The standardized test statistic is
The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances and are equal.
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26. Two Sample t-Test for the Difference Between Means
Variances are equal
Information from the two samples is combined to calculate a pooled estimate of the standard deviation .
The standard error for the sampling distribution of is
d.f.= n1 + n2 – 2
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27. Two Sample t-Test for the Difference Between Means
Variances are not equal
If the population variances are not equal, then the standard error is
d.f = smaller of n1 – 1 and n2 – 1
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28. Normal or t-Distribution?
Are both sample sizes at least 30?
Yes
Use the z-test. No
You cannot use the z-test or the t-test.
Are both populations normally distributed? No
Use the t-test with
Yes
Are the population variances equal?
Are both population standard deviations known? No
Yes
d.f = n1 + n2 – 2. No
Yes
Use the t-test with
d.f = smaller of n1 – 1 or n2 – 1.
Use the z-test.
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29. Two-Sample t-Test for the Difference Between Means (Small Independent Samples)
In Words In Symbols
State the claim mathematically and verbally. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the degrees of freedom.
Determine the critical value(s).
State H0 and Ha.
Identify α.
d.f. = n1+ n2 – 2 or
d.f. = smaller of n1 – 1 or n2 – 1.
Use Table 5 in Appendix B.
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30. Two-Sample t-Test for the Difference Between Means (Small Independent Samples)
In Words In Symbols
Determine the rejection region(s).
Find the standardized test statistic and sketch the sampling distribution.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If t is in the rejection region, reject H0. Otherwise, fail to reject H0.
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31. Example: Two-Sample t-Test for the Difference Between Means
The results of a state mathematics test for random samples of students taught by two different teachers at the same school are shown below. Can you conclude that there is a difference in the mean mathematics test scores for the students of the two teachers? Use α = Assume the populations are normally distributed and the population variances are not equal.
Teacher 1
Teacher 2
s1 = 39.7
s2 = 24.5
n1 = 8
n2 = 18
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32. Solution: Two-Sample t-Test for the Difference Between Means
Test Statistic:
H0:
Ha:
α =
d.f. =
Rejection Region:
μ1 = μ2
μ1 ≠ μ2 (claim)
0.10
8 – 1 = 7
Decision:
Fail to Reject H0 .
At the 10% level of significance, there is not enough evidence to support the claim that the mean mathematics test scores for the students of the two teachers are different.
t ≈ 0.922
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33. Example: Two-Sample t-Test for the Difference Between Means
A manufacturer claims that the mean calling range (in feet) of its 2.4-GHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected similar phones from its competitor. The results are shown below. At α = 0.05, can you support the manufacturer’s claim? Assume the populations are normally distributed and the population variances are equal.
Manufacturer
Competitor
s1 = 45 ft
s2 = 30 ft
n1 = 14
n2 = 16
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34. Solution: Two-Sample t-Test for the Difference Between Means
Ha:
α =
d.f. =
Rejection Region:
μ1 ≤ μ2
μ1 > μ2 (claim)
0.05
– 2 = 28 t
1.701
0.05
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35. Solution: Two-Sample t-Test for the Difference Between Means
Test Statistic:
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36. Solution: Two-Sample t-Test for the Difference Between Means
Ha:
α =
d.f. =
Rejection Region:
μ1 ≤ μ2
μ1 > μ2 (claim)
Test Statistic:
0.05
– 2 = 28
Decision:
Reject H0 .
At the 5% level of significance, there is enough evidence to support the manufacturer’s claim that its phone has a greater calling range than its competitors. t
1.701
0.05
1.811
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37. Section 8.2 Summary
Performed a t-test for the difference between two means μ1 and μ2 using small independent samples
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38. Testing the Difference Between Means (Dependent Samples)
Section 8.3
Testing the Difference Between Means (Dependent Samples)
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39. Section 8.3 Objectives
Perform a t-test to test the mean of the differences for a population of paired data
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40. t-Test for the Difference Between Means
To perform a two-sample hypothesis test with dependent samples, the difference between each data pair is first found:
d = x1 – x2 Difference between entries for a data pair
The test statistic is the mean of these differences.
Mean of the differences between paired data entries in the dependent samples
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41. t-Test for the Difference Between Means
Three conditions are required to conduct the test.
The samples must be randomly selected.
The samples must be dependent (paired).
Both populations must be normally distributed.
If these requirements are met, then the sampling distribution for is approximated by a t-distribution with n – 1 degrees of freedom, where n is the number of data pairs.
-t0 t0 μd
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42. Symbols used for the t-Test for μd
Symbol Description n
The number of pairs of data d
The difference between entries for a data pair, d = x1 – x2
The hypothesized mean of the differences of paired data in the population
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43. Symbols used for the t-Test for μd
Symbol Description
The mean of the differences between the paired data entries in the dependent samples sd
The standard deviation of the differences between the paired data entries in the dependent samples
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44. t-Test for the Difference Between Means
The test statistic is
The standardized test statistic is
The degrees of freedom are
d.f. = n – 1.
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45. t-Test for the Difference Between Means (Dependent Samples)
In Words In Symbols
State the claim mathematically and verbally. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the degrees of freedom.
Determine the critical value(s).
State H0 and Ha.
Identify α.
d.f. = n – 1
Use Table 5 in Appendix B if n > 29 use the last row (∞) .
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46. t-Test for the Difference Between Means (Dependent Samples)
In Words In Symbols
Determine the rejection region(s).
Calculate and
Find the standardized test statistic.
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47. t-Test for the Difference Between Means (Dependent Samples)
In Words In Symbols
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If t is in the rejection region, reject H0. Otherwise, fail to reject H0.
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48. Example: t-Test for the Difference Between Means
A shoe manufacturer claims that athletes can increase their vertical jump heights using the manufacturer’s new Strength Shoes®. The vertical jump heights of eight randomly selected athletes are measured. After the athletes have used the Strength Shoes® for 8 months, their vertical jump heights are measured again. The vertical jump heights (in inches) for each athlete are shown in the table. Assuming the vertical jump heights are normally distributed, is there enough evidence to support the manufacturer’s claim at α = 0.10? (Adapted from Coaches Sports Publishing)
Athletes 1 2 3 4 5 6 7 8
Height (old) 24 22 25 28 35 32 30 27
Height (new) 26 29 33 34
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49. Solution: Two-Sample t-Test for the Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
H0:
Ha:
α =
d.f. =
Rejection Region:
μd ≥ 0
μd < 0 (claim)
0.10
8 – 1 = 7
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50. Solution: Two-Sample t-Test for the Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
Before
After d d2 24 26 –2 4 22 25 –3 9 28 29 –1 1 35 33 2 32 34 30 –5 27
Σ = –14
Σ = 56
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51. Solution: Two-Sample t-Test for the Difference Between Means
d = (jump height before shoes) – (jump height after shoes)
H0:
Ha:
α =
d.f. =
Rejection Region:
Test Statistic:
µd ≥ 0
μd < 0 (claim)
0.10
8 – 1 = 7
Decision:
Reject H0.
At the 10% level of significance, there is enough evidence to support the shoe manufacturer’s claim that athletes can increase their vertical jump heights using the new Strength Shoes®.
t ≈ –2.333
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52. Section 8.3 Summary
Performed a t-test to test the mean of the difference for a population of paired data
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53. Testing the Difference Between Proportions
Section 8.4
Testing the Difference Between Proportions
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54. Section 8.4 Objectives
Perform a z-test for the difference between two population proportions p1 and p2
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55. Two-Sample z-Test for Proportions
Used to test the difference between two population proportions, p1 and p2.
Three conditions are required to conduct the test.
The samples must be randomly selected.
The samples must be independent.
The samples must be large enough to use a normal sampling distribution. That is, n1p1 ≥ 5, n1q1 ≥ 5, n2p2 ≥ 5, and n2q2 ≥ 5.
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56. Two-Sample z-Test for the Difference Between Proportions
If these conditions are met, then the sampling distribution for is a normal distribution.
Mean:
A weighted estimate of p1 and p2 can be found by using
Standard error:
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57. Two-Sample z-Test for the Difference Between Proportions
The test statistic is
The standardized test statistic is
where
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58. Two-Sample z-Test for the Difference Between Proportions
In Words In Symbols
State the claim. Identify the null and alternative hypotheses.
Specify the level of significance.
Determine the critical value(s).
Determine the rejection region(s).
Find the weighted estimate of p1 and p2. Verify that
State H0 and Ha.
Identify α.
Use Table 4 in Appendix B.
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59. Two-Sample z-Test for the Difference Between Proportions
In Words In Symbols
Find the standardized test statistic and sketch the sampling distribution.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If z is in the rejection region, reject H0. Otherwise, fail to reject H0.
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60. Example: Two-Sample z-Test for the Difference Between Proportions
In a study of 150 randomly selected occupants in passenger cars and 200 randomly selected occupants in pickup trucks, 86% of occupants in passenger cars and 74% of occupants in pickup trucks wear seat belts. At α = 0.10 can you reject the claim that the proportion of occupants who wear seat belts is the same for passenger cars and pickup trucks? (Adapted from National Highway Traffic Safety Administration)
Solution:
1 = Passenger cars 2 = Pickup trucks
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61. Solution: Two-Sample z-Test for the Difference Between Means
Ha:
α =
n1= , n2 =
Rejection Region:
p1 = p2 (claim)
p1 ≠ p2
0.10
150
200
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62. Solution: Two-Sample z-Test for the Difference Between Means
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63. Solution: Two-Sample z-Test for the Difference Between Means
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64. Solution: Two-Sample z-Test for the Difference Between Means
Ha:
α =
n1= , n2 =
Rejection Region:
p1 = p2
p1 ≠ p2
Test Statistic:
0.10
150
200
Decision:
Reject H0
At the 10% level of significance, there is enough evidence to reject the claim that the proportion of occupants who wear seat belts is the same for passenger cars and pickup trucks.
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65. Example: Two-Sample z-Test for the Difference Between Proportions
A medical research team conducted a study to test the effect of a cholesterol-reducing medication. At the end of the study, the researchers found that of the 4700 randomly selected subjects who took the medication, 301 died of heart disease. Of the 4300 randomly selected subjects who took a placebo, 357 died of heart disease. At α = 0.01 can you conclude that the death rate due to heart disease is lower for those who took the medication than for those who took the placebo? (Adapted from New England Journal of Medicine)
Solution:
1 = Medication 2 = Placebo
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66. Solution: Two-Sample z-Test for the Difference Between Means
Ha:
α =
n1= , n2 =
Rejection Region:
p1 ≥ p2
p1 < p2 (claim)
0.01
4700
4300 z
–2.33
0.01
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67. Solution: Two-Sample z-Test for the Difference Between Means
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68. Solution: Two-Sample z-Test for the Difference Between Means
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69. Solution: Two-Sample z-Test for the Difference Between Means
Ha:
α =
n1= , n2 =
Rejection Region:
p1 ≥ p2
p1 < p2 (claim)
Test Statistic:
0.01
4700
4300
Decision:
Reject H0
At the 1% level of significance, there is enough evidence to conclude that the death rate due to heart disease is lower for those who took the medication than for those who took the placebo. z
–2.33
0.01
–2.33
–3.46
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70. Section 8.4 Summary
Performed a z-test for the difference between two population proportions p1 and p2
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