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Answers for Project 3
Minitab output for 95% confidence interval for p = the true proportion of all truck drivers who suffer from sleep apnea (Exercise 7.45, McClaveâs text):
Test and Confidence Interval for One Proportiom
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 95.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.715473, 0.844276) 7.06 0.
QUESTION 1: Based on the data in Exercise 7.45 of McClaveâs text, what is the estimate for the proportion of all truck drivers who suffer from obstructive sleep apnea?
- On the output, the value under âSample pâ is the estimator for the true population proportion, p. Thus, Ëp = .779874 is the estimator for the true proportion of all truck drivers who suffer from sleep apnea.
QUESTION 2: What is a 95% confidence interval for the proportion of all truck drivers who suffer from obstructive sleep apnea?
- On the output, the interval under â95.0% CIâ is the 95% confidence interval for the true population proportion, p. Thus, 95% confidence interval for the true proportion of all truck drivers who suffer from sleep apnea is (0. 715473 , 0 .844276).
QUESTION 3: What is a 90% confidence interval for the proportion of all truck drivers who suffer from obstructive sleep apnea? How does this interval compare to the interval that you obtained in your answer to question 2?
Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 90.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.725827, 0.833922) 7.06 0.
- Thus, 90% confidence interval for the true proportion of all truck drivers who suffer from sleep apnea is (0. 725827 , 0 .833922).
- As expected, the 90% confidence interval is shorter than the 95% interval obtained as the answer to Question 1.
QUESTION 4: Among the intervals 85%, 90%, 95% 96%, 98% and 99% that you obtained, which is the shortest? Which is the longest? Are these results surprising to you? Explain.
Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 85.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.732573, 0.827175) 7.06 0.
Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 96.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.712391, 0.847358) 7.06 0.
Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 98.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.703434, 0.856315) 7.06 0.
Test and Confidence Interval for One Proportion
Test of p = 0.5 vs p not = 0.
Sample X N Sample p 99.0 % CI Z-Value P-Value 1 124 159 0.779874 (0.695236, 0.864512) 7.06 0.
- The 85% confidence interval, (0. 732573 , 0 .827175), is the shortest.
- The 99% confidence interval, (0. 695236 , 0 .864512), is the longestest.
- No surprise. When the same data is used to construct confidence intervals with different confidence coefficients, intervals with higher confidence coefficients will always be longer than intervals with lower confidence coefficients. This is because the multiplier of the standard error of the estimator will be larger for larger confidence coefficients.
QUESTION 10: What is a 95% confidence interval for the proportion of female honors STA 2023 students who are âsurvey respondersâ?
- The 95% confidence interval for the proportion of female honors students who are âsurvey responders, (0. 352464 , 0 .674563).
QUESTION 11: How does the upper endpoint of the interval that you obtained in Question 8 compare to the lower endpoint of the interval that you obtained in Question 10? How does the lower endpoint of the interval that you obtained in Question 8 compare to the upper endpoint of the interval that you obtained in Question 10? Explain.
- Upper endpoint from Question 8 is 0.647536. Lower endpoint from Question 10 is 0 .352464. Notice that 0.647536 = 1 â 0 .352464.
- Lower endpoint from Question 8 is 0.325437. Upper endpoint from Question 10 is 0 .674563. Notice that 0.325437 = 1 â 0 .674563.
- (Proportion of responders who are male) + (Proportion of responders who are female) = 1!! If the largest plausible value (at the 95% confidence level) for the proportion of repon- ders whe are males is 0.647536, then the smallest plausible value (at the 95% confidence level) for the proportion of responders who are female is 1â 0 .647536 = 0.352464, exactly the value that we obtained when we directly computed the 95% confidence interval for the proportion of female responders. enditemize
QUESTION 12: What are the appropriate research and null hypotheses in Exercise 8.64 in McClaveâs text.?
- If p = true proportion of all women over age 40 who will experience skin improvements when using alpha-hydroxy acid:
Ha : p >. 60 Ho : p =. 60.
QUESTION 13: Based on the data in Exercise 8.64 of McClaveâs text, what is the estimate for the proportion of all women over age 40 who exhibit skin improvement using the cream with alpha-hydoxy acid?
Test and Confidence Interval for One Proportion
Test of p = 0.6 vs p > 0.
Sample X N Sample p 95.0 % CI Z-Value P-Value 1 23 33 0.696970 (0.540171, 0.853768) 1.14 0.
- Estimate for p is Ëp = .0696970.
QUESTION 14: What is the calculated value of the zâstatistic for testing the hypotheses you gave in Question 12?
- See the Minitab output in the answer to Question 13. z = 1.14.
QUESTION 15: What is the pâvalue for testing the hypotheses you gave in Question 12?
- See the Minitab output in the answer to Question 13. p â value = 0.128.
QUESTION 16: If you were interested in testing the hypotheses from Question 12 at the α = .05 level of significance, what would you conclude? Be sure to phrase your conclusion in terms of this problem.
- α = .05 is smaller than the pâvalue = 0.128. The null hypothesis can be rejected for any value of α â„ 0 .128, but not for any value of α smaller than 0.128. So, in this case, at the α = .05 level we CANNOT conclude that the cream will improve the skin of more than 60% of the women over age 40.
