1. All confidence intervals have the form: Hint: The word "estimate" refers to "the sample mean ". See the box "Confidence Interval" on p378. a. estimate + * times standard error of X. b. estimate + * times margin of error c. estimate + margin of error. d. Both a and c. 2. Which of the following is an example of statistical inference? a. Calculating a confidence interval b. Calculating a standard deviation c. Conducting a hypothesis test d. Both a and c 3. A confidence interval is used to estimate a. the shape of the population's distribution b. the sampling distribution of c. a sample statistic x d. a population parameter 4. The standard error of the sample mean is Hint: See the #8 in Ch 16 & 20 one-sample t test.5. The standard error measures the average distance between and the Therefore, it is desirable to have a standard error as as possible. Hint: Recall that the standard error = standard deviation of all sample means of the same size (with " replaced by s). Review Ch 15 if you are unclear about the concept of the standard deviation of the sampling distribution of x. a. a sample mean, population mean, small b. a sample mean, population mean, large C . an individual observation in a sample, sample mean, small d. an individual observation in a population, population mean, large 6. A simple random sample with n = 54 provided a sample mean of 22.5 and a sample standard deviation of 4.4. With 99% confidence, what is * critical value for the estimation of the population mean? Hint: Use the nearest degrees of freedom: df = 50 a. 2.009 b. 2.123 c. 2.678 d. 2.937 7. Refer to the question above. What is the margin of error for the estimation of the population mean? a. 0.3241 b. 1.6034 c. 2.3876 d. 2.9940 8. Refer to the question above. What is the 99% confidence interval for the estimation of the population mean? a. (14.22, 23.15) b. (15.31, 23.88) C. (20.89, 24.10) d. (22.11, 27.57) 9. A college admissions director wants to estimate the average age of students in their graduate programs. Assume that the age of graduate students is Normally distributed with mean /. A random sample of 7 graduate students was selected and their ages are: 25, 36, 27, 29, 30, 34, 25. The standard deviation of the sample is 4.2761. Based on these data, the margin of error for a 90% confidence interval for / based on these data is . Consequently, the 90% confidence interval for u is a. 1.6162, 29.43 + 1.6162 b. 2.8711, 29.43 # 2.8771 c. 3.1402, 29.43 # 3.1403 d. 3.0075, 29.43 # 3.997510. Let x = 10. We are interested in a 95% confidence interval. If the margin of error is 8, then u lies between and For the same sample mean, if the margin of error is 3, then u lies between and a. 1, 19, 3. 12 b. 2, 10, 6, 9 c. 2, 18, 7, 13 d. 4. 13, 8 11 11. Mathematical fact: Refer to the question above. A smaller margin of error will lead to a confidence interval, resulting in a precise estimate of u (because a narrow confidence interval allows us to pin down / more precisely.) Note: That's why it is desirable to have a small margin of error! a. wider, more wider, less C. narrower, more d. narrower, less - 5 - E. Confidence Level We would like to have high confidence on our estimate of u. However, there is a trade-off. The next several questions explore what this trade-off is. See also 4/29 (Lab work). 12. Suppose n = 11. What is the critical value (*) for a 80% confidence level? a. 0.703 b. 0.865 c. 0.988 d. 1.372 13. Assuming that the sample size is the same as given above (n = 11), what is the critical value (*) for a 98% confidence level? a. 1.760 b. 1.988 c. 2.127 d. 2.764 14. If we increase the confidence level, the critical value (*) becomes Hint: Compare your answers of the questions above. See also the diagrams and compare the positions of the two critical values (1*). a. farther away from the center of 0 8 b. larger c. smaller . + d. Both a and b 0-98 15. A sample of 11 items resulted in a sample mean of 30 with standard deviation 5. The margin of error for a 80% confidence interval is and the margin of error for a 98% confidence interval is a. 2.068, 4.166 b. 4.777, 2.068 c. 2.321, 3.899 d. 3.987, 4.12316. Mathematical fact The higher the confidence level, the the margin of error; consequently, the the confidence interval and the precise our estimate of the population will be. Hint: You can see the trade-off here. On one hand, we want high confidence. But on the other hand, high confidence leads to wider confidence intervals, which result in less precise estimates of u because narrow confidence intervals allow us to pin down u more precisely. No free lunch in this world indeed! a. smaller, wider, more b. larger, wider, less c. smaller, narrower, less d. larger, narrower, more - 7 - F. Sample Size Remember our goal: a small margin of error so that we can obtain a narrow confidence interval to pin down a more precisely. In the next several questions, you will find out how sample size affects the margin of error via: the standard error the critical value (*). F.1 - Standard Error 17. A sample of 64 items resulted in a sample mean of 25 and standard deviation of 15. The standard error of sample mean is a. 1.766 c. 2.091 b. 1.875 d. 2.433 18. Refer to the question above. Assume that the sample mean and standard deviation remain unchanged. Suppose the sample size is 100 instead of 64. Then the standard error of sample mean is Note: We have no control over the sample mean or the sample standard deviation. Just think of them as fixed. In other words, we hold them "constant" while changing the sample size n so that we can see the role played by n in our calculation more easily. a. 1.2 C. 2.9 b. 1.5 d. 3.2[9. Mathematical Fact Compare the hurt questions above. Assume that the sample mean and sample standard deviation are held constant (tint is. remain the same}. The linger the sample size. the the standard error: consequently. the the margin of error. 5 s Hint: [falli becomes smaller. then the expression I" x will also become smaller. See the formula below. t i x i r J; I'_l margin of error a. smaller. larger b. larger. smaller c. larger. larger d. smaller. smaller [-23 Critical value If\"! Not only does sample size affect the standard error. but also the critical value L"). 20. Let the condence level be 90%. ll' n = 25.1"I is . Il'n = 100.." is a. 3.l I I. 290 b. 2.931. 3.323 c. I331. [.221 cl. I.':' I I. [.660 2|. Mathematical Fact The larger the sample sire. the the critical value {1"} and consequently. the the margin ol'error. I Hint: If r' becomes smaller. then the expression r\" \23. Refer to the cholesterol above. Suppose the researcher gave the drug to 100 instead of 25 subjects and let's assume that the sample mean and the standard deviation remain the same (that is, x = 90 and s = 30). Then a 90% confidence interval for u is mg/dl. a. 90 + 4.76 b. 90 + 4.98 C. 90 # 5.01 d. 90 + 5.32 24. Mathematical Fact Compare the two questions above. A larger sample size would produce a margin of error and consequently, a confidence interval. smaller, narrower smaller, wider C. larger, narrower larger, narrower. 25. You plan to construct a confidence interval for the mean of a Normal population. Which of the following will reduce the size of the margin of error? Using a lower level of confidence Increasing the sample size Use a larger critical value Both a and b.26. A 95% confidence interval has the following interpretation: Hint: See the box "Interpreting a Confidence Interval" on p379 and Fig 16.2 on p380. If we repeat our study 100 times, we expect (or hope) that our sample mean x is one of those ("lucky") out of 100 sample means that stays close to the population mean. ("Stays close" in this case means our x stays within the "+ margin-of-error- neighborhood" of u.) If so, then our estimate for u will be To the contrary, if our sample mean is one of the "unlucky" ones that stays far from u ("far" means our x lies outside the "+ margin-of-error-neighborhood of u), then our estimate will be Do we know for sure whether our sample mean is close to or far from u? a. 95, correct, 5, incorrect, No, we do not know b. 95, incorrect, 6, correct, Yes, we always know c. 82, incorrect, 18, correct, Yes, we always know d. 82, correct, 18, incorrect No, we do not know 27. We can think of a confidence level as a success rate of our method. For example, a 99% confidence level means that we expect our estimate for / to be accurate roughly out of 100 times. Hint: See the box "Interpreting a Confidence Interval" on p379 and Fig 16.2 on p380. a. 100 b. 99 C. 98 d. 05 -11 - 28. Clinical literature reports that the duration of a typical cold is roughly 18 days. Researchers wanted to know if people tend to underestimate the duration of a typical cold, on average. They surveyed a random sample of 352 healthy adults in Georgia and asked them how long they think (expect) that a typical cold lasts. The researchers reported a 95% confidence interval of 6.9 to 8.2 days for the mean expected duration of typical cold. Based on the confidence interval, do healthy adults in Georgia underestimate how long a typical cold lasts on average? Hint: p = the average number of days people think (or expect) a typical cold lasts. According to the survey, u is estimated to lie within 6.9 to 8.2 days. In other words, people think (expect) a typical cold lasts between 6.9 to 8.2 days. a. Yes, because nobody in the survey answered 18. b. No, because not all healthy adults in Georgia participated. c. Yes, because the value 18 is greater than the values in the confidence interval. d. No, because the value 18 is greater than the values in the confidence interval. 29. A researcher read from a science magazine that the number of seeds per fruit for a certain freshwater plant is about 400. The researcher decided to find out if this claim is true by collecting a sample and obtained a confidence interval of (277 seeds per fruit, 432 seeds per fruit). Based on this confidence interval, does it appear that the population mean number of seeds per fruit for that particular freshwater plant is different from the mean reported by the science magazine? a. Yes, the estimate of the population mean from the confidence interval is different than the number reported by the science magazine because 400 is included in the confidence interval. b. No, the estimate of the population mean from the confidence interval is not different than the number reported by the science magazine because 400 is included in the confidence interval