STA 2023
Chapter 9 – Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses
- Comparing Two Population Means: Independent Sampling (9.1)
- Sampling Distribution of the Difference Between Two Sample Means
- will be normal when n1 30 and n2 30
- Example – Distribution of the Difference Between Two Sample Means
- Suppose 1 = 12 and 1 = 2 while 2 = 10 and 2 = 4. Consider drawing random samples of size 36 from population 1 and random samples of size 64 from population 2. Draw the sampling distribution of . 2, and .6009, so the distribution will be normally distributed centered at the mean = 2, and going left to three standard deviations below the mean at .1973 and going right to three standard deviations above the mean at 3.8027.
- Find P(>3.5). Since the distribution of is approximately normal by the Central Limit Theorem, then we can convert to standard normal as follows: 2.50, and P(z > 2.50) = .0062.
- Example – Chapter 9 Examples
- Question 1: (6.2, 25.8). We are 95% confident that the difference between the average lifetime of GE light bulbs and the average lifetime of Sylvania light bulbs is between 6.2 hours and 25.8 hours. Since this interval does not include 0, we can conclude that one brand is usually better than the other – specifically, the GE light bulbs.
- Question 2: .2, =18, (-5.35, -4.65). We are 90% confident that the difference between the average nicotine content of Virginia Slims and the average nicotine content of Jacks cigarettes is between –5.35 mg and –4.65 mg. Since this interval does not include 0, we can conclude that one brand contains more nicotine content than the other – specifically, the Jacks cigarettes.
- Question 3:
Hypotheses: H0: 1 - 2 = 0, Ha: 1 - 2 > 0
Rejection Region: Since =.01 and v=6 we should use t > 3.143 as our rejection region
Test Statistic: 762.5,2.56
P-value: Since our rejection is right-tailed, we should find the area under the curve to the right of 2.56. From Table VI, P(t6 > 2.56) is between .01 and .025, thus .01 < p-value < .025
Conclusion: Do not reject H0. There is not enough evidence to suggest that the Benjamin Moore paints covers a larger area on average than the Pittsburgh paints, at =.01.
- Question 4:
Hypotheses: H0: 1 - 2 = 0, Ha: 1 - 2 0
Rejection Region: Since =.05 we should use z > 1.96 or z < -1.96 as our rejection region
Test Statistic: 2.24
P-value: Since our rejection is two-tailed, we should find the area under the curve to the right of 2.24, and double it. From Table IV, P(z > 2.24) = .0125, so p-value = .0250
Conclusion: Reject H0. There is sufficient evidence to suggest that the cephalic indices of the island inhabitants are significantly different, and they are not of the same racial ancestry, at =.05.
- Comparing Two Population Means: Paired Difference Experiments (9.2)
- Example – Chapter 9 Examples
- Question 5a: (.58, 1.92). We are 80% confident that true mean difference between the number of accidents before and after the new traffic-control system is between .58 and 1.92. There is a significant difference in the number of accidents.
- Question 5b:
Hypotheses: H0: D = 0, Ha: D 0
Rejection Region: Since =.02 we should use t < -2.718 or t > 2.718 as our rejection region
Test Statistic: 2.53
P-value: Since our rejection region is two-tailed, we should find the area under the curve to the right of 2.53, and double it. From Table VI, P(t11 > 2.53) is between .01 and .025, so .02 < p-value < .05
Conclusion: Do not reject H0. There is not enough evidence to suggest that the new traffic-control system has a significant effect on the number of accidents, at =.02.
- Question 6a: (10.22, 12.18). We are 95% confident that the true mean difference of reduction in systolic blood pressure is between 10.22 bpm and 12.18 bpm.
- Question 6b:
Hypotheses: H0: D = 10, Ha: D > 10
Rejection Region: Since =.05 we should use z > 1.645 as our rejection region
Test Statistic: 2.4
P-value: Since our rejection region is upper-tailed, we should find the area to the right of 2.4. From Table IV, P(z > 2.4) = .0082, so our p-value = .0082
Conclusion: Reject H0. There is sufficient evidence to suggest that the true mean difference of reduction in systolic blood pressure is greater than 10 beats per minute, at =.05.
- Comparing Two Population Proportions: Independent Sampling (9.3)
- Example – Chapter 9 Examples
- Question 7a:(.63, .69) (0, 1) (.56, .64) (0, 1) Since both intervals fall between 0 and 1, the sample size is sufficiently large
- Question 7b: .06.13 = (-.07, .19). We are 99% confident that the difference between the true proportions of male and female voters who favor the candidate is between -.07 and .19. Since 0 is included in this interval, we cannot conclude that one sex favors the candidate more.
- Question 7c:
Hypotheses: H0: (p1 – p2) = 0, Ha: (p1 – p2) < 0
Rejection Region: Since =.10 we should use z > 1.28 as our rejection region
Test Statistic: .634, so 1.15
P-value: Since our rejection region is lower-tailed, we should find the area to the left of 1.15. From Table IV, P(z < 1.15) = .8749, so our p-value is .8749
Conclusion: Do not reject H0. There is not enough evidence to suggest that the proportion of males who favor the candidate is larger than the proportion of females who favor the candidate, at =.10.
- Determining Sample Size (9.4)
- Example – Chapter 9 Examples
- Question 8: 50, so 135.3 136
- Question 9: .03, 2134.2 2135
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