THE DIVERSITY OF SAMPLES FROM THE SAME POPULATION

Transcription

1 CHAPTER 19 THE DIVERSITY OF SAMPLES FROM THE SAME POPULATION Narrative: Bananas Suppose a researcher asks the question: What is the average weight of bananas selected for purchase by customers in grocery stores? Assume the distribution of weights is the same across all stores. 1. {Bananas narrative} How can the researcher go about answering this question? Give a general description of the process. ANSWER: TAKE A SAMPLE, RECORD THE AVERAGE WEIGHT OF THE SAMPLE, AND THEN USE THAT TO ANSWER THE QUESTION ABOUT THE POPULATION (ALONG WITH SOME MEASURE OF ERROR). 2. {Bananas narrative} Suppose the researcher takes a sample of 100 shoppers and finds the average weight of their banana purchases to be 2.1 pounds. Can he just go ahead and report that the average banana purchase for all grocery store customers is 2.1 pounds? Why or why not? ANSWER: NO; SAMPLE RESULTS WILL VARY FROM SAMPLE TO SAMPLE, SO THE SAMPLE AVERAGE WILL PROBABLY NOT BE EXACTLY EQUAL TO THE POPULATION AVERAGE. 3. Which of the following statements is false? a. Sample results will always be very close to their respective population values. b. Sample results vary from one sample to the next. c. The key to interpreting statistical results is to understand what kind of dissimilarity we should expect to see in various samples from the same population. d. None of the above statements are false. 4. Suppose you knew that most samples were likely to provide an answer that is within 10% of the population value. What would also be true in that case? a. The population value should be within 10% of whatever our specific sample gave us. b. 10% of the population values should be close to whatever our specific sample gave us. c. The chance that a specific sample answer is correct (equal to the population value) is 90%. d. All of the above statements are true. Narrative: Politics Suppose a population contains 60% Republicans and 40% Democrats. 5. {Politics narrative} Suppose you take a random sample of 10 people from this population. Are you certain that you would get 6 Republicans and 4 Democrats in your sample? Explain your answer. ANSWER: NO; SAMPLE RESULTS VARY FROM SAMPLE TO SAMPLE; OR, THE LONG-TERM AND SHORT-TERM PROBABILITIES ARE NOT THE SAME.

2 6. {Politics narrative} Is it possible to get a random sample that does not represent the population well, in terms of Democrats and Republicans? Explain your answer. ANSWER: YES; THIS CAN HAPPEN JUST BY CHANCE, BECAUSE SAMPLE RESULTS VARY, ESPECIALLY WITH SMALL SAMPLES. 7. {Politics narrative} Suppose you take a random sample of 10 people from this population. Does the rule for sample proportions apply in this situation? Explain your answer. ANSWER: NO; THE SAMPLE SIZE IS NOT LARGE ENOUGH. THE EXPECTED NUMBER OF DEMOCRATS IS ONLY 4, WHICH IS LESS THAN THE 5 NEEDED FOR THE RULE TO HOLD. 8. {Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population. How will the shape, mean, and standard deviation of the frequency curve for the proportions of Democrats in the samples differ from the shape, mean, and standard deviation of the frequency curve for the proportions of Republicans in the samples? ANSWER: BOTH WILL BE BELL-SHAPED; BOTH WILL HAVE THE SAME STANDARD DEVIATION (.015); THE MEANS WILL DIFFER (.60 FOR THE REPUBLICANS, AND.40 FOR THE DEMOCRATS). 9. In practice you don t know the population value, and you take a sample in order to estimate what the population value is. Once you take a specific sample, is it possible to determine whether or not that sample is an accurate reflection of the population? Explain your answer. ANSWER: NO; YOU WOULD HAVE TO KNOW THE POPULATION VALUE, AND YOU DON T, WHICH WAS THE WHOLE POINT OF TAKING THE SAMPLE. EVEN RANDOM SAMPLES CAN BE MISREPRESENTATIVE, JUST BY CHANCE. Narrative: cell phone owners Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners. 10. {Cell phone owners narrative} The frequency curve made from proportions of cell phone owners from the various samples of size 2,500 from this population will have what approximate shape? ANSWER: BELL-SHAPED CURVE 11. {Cell phone owners narrative} The frequency curve made from proportions of cell phone owners from the various samples of size 2,500 from this population will have what approximate mean and standard deviation? ANSWER: MEAN:.20 OR 20%; STANDARD DEVIATION:.008 OR.8%. 12. {Cell phone owners narrative} What is the chance that a sample of size 2,500 from this population will contain at least 20% cell phone owners? ANSWER: 50% OR {Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this population and found that 17.6% of them owned a cell phone. Is this considered to be a reasonable value given the size of this sample? Use the standardized score in your answer. ANSWER: THE STANDARDIZED SCORE IS -3. THIS IS NOT WITHIN THE REASONABLE EXPECTATIONS FOR A SAMPLE OF THIS SIZE. 14. {Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this population and found that 21.6% of the people in this sample owned a cell phone. Is this considered to be a reasonable value given the size of this sample? Use the standardized score in your answer. ANSWER: THE STANDARDIZED SCORE IS +2. THIS IS WITHIN THE REASONABLE EXPECTATIONS FOR A SAMPLE OF THIS SIZE, BUT IS ON THE MARGIN.

3 15. {Cell phone owners narrative} What is the chance that less than 20.8% of the people in a sample of size 2,500 from this population will own a cell phone? ANSWER: 84% OR.84. (THE STANDARDIZED SCORE IS 1.) 16. {Cell phone owners narrative} What is the chance that the proportion of cell phone owners in a sample of size 2,500 from this population will be more than two standard deviations from the expected mean? ANSWER: 5% OR {Cell phone owners narrative} What range of proportions of cell phone owners is reasonable to expect from this population (assuming your sample size is 2,500)? Justify your answer ANSWER: 95% OF THE SAMPLE PROPORTIONS SHOULD LIE BETWEEN.184 AND {Cell phone owners narrative} Suppose your sample size was only 250. What range of proportions of cell phone owners is reasonable to expect from this population? Justify your answer ANSWER: 95% OF THE SAMPLE PROPORTIONS WILL LIE BETWEEN.149 AND {Cell phone owners narrative} How would the frequency curve made from proportions of cell phone owners from the various samples of size 2,500 compare to the frequency curve made from proportions of cell phone owners from the various samples of size 250? ANSWER: BOTH WILL BE BELL-SHAPED; BOTH WILL HAVE A MEAN OF.20 OR 20%; THE STANDARD DEVIATION FOR SAMPLES OF SIZE 2,500 IS.008 OR 0.8%, WHILE THE STANDARD DEVIATION FOR SAMPLES OF SIZE 250 IS.025 OR 2.5%. 20. In which of the following situations does the rule for sample proportions apply? a. A pollster takes a random sample of 1,000 Americans and asks their opinion on the President (approve/disapprove/neutral). He is interested in the percentage who approve of the President. b. You want to know whether or not people like the new CD by your favorite artist. You ask 5 people and record the percentage who say they like it. c. A researcher weighs the same newborn baby each week for one year, and records whether or not the child is within the normal weight range. At the end of the year, he records the percentage of times that the child was within the normal weight range. d. All of the above. 21. If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have what approximate shape? a. A bar graph with two bars, one for the proportion having the trait of interest, and the other for the proportion not having the trait of interest. b. A bell-shape. c. A flat shape; each outcome should be equally likely. d. Unknown; it can change every time. ANSWER: B 22. If numerous large random samples or repetitions of the same size are taken from a population, the proportions from the various samples will have what approximate mean? a. The true population proportion. b. The true population average. c. 95% because most of them will be within 2 standard deviations of the true population value. d. None of the above.

4 23. {Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population. The proportions of Democrats from the various samples of size 1,000 will have what approximate standard deviation? a..24 or 24% b or.024% c..015 or 1.5% d. None of the above. ANSWER: C 24. {Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population. The proportions of Republicans from the various samples of size 1,000 will have what approximate standard deviation? a..24 or 24% b or.024% c..015 or 1.5% d. None of the above. ANSWER: C 25. If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have an approximate shape. ANSWER: BELL 26. If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have a mean that is the true population proportion. ANSWER: EQUAL TO 27. The standard deviation of the proportions from numerous random samples of size 1,000 from a population will be the standard deviation of the proportions from numerous random samples of size 10,000 from the same population. ANSWER: GREATER THAN Narrative: Test scores Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve. 28. {Test scores narrative} Suppose you take numerous random samples of size 100 from this population. Describe the shape and give the mean and standard deviation of the resulting frequency curve. ANSWER: BELL-SHAPED; MEAN=77; STANDARD DEVIATION= {Test scores narrative} Suppose you take a single random sample of size 100 people from this population. What is the chance that their average test score will be above 77? ANSWER: 50% OR {Test scores narrative} Suppose you take a single random sample of size 100 from this population, and you get a mean test score of 76. Is this something that you would have expected? Use a probability to justify your answer. ANSWER: 2.5% OR.025 OF THE MEAN TEST SCORES FOR A SAMPLE OF THIS SIZE WILL LIE AT OR BELOW 76 (OR, THE STANDARDIZED SCORE IS -2). THIS IS ON THE BORDERLINE OF WHAT WE WOULD EXPECT FOR THIS POPULATION.

5 31. {Test scores narrative} Suppose you take a single random sample of size 100 from this population, and you get a mean test score of 79. Is this something that you would have expected? Use a probability to justify your answer. ANSWER: NO. THE STANDARD SCORE IS +4. SAMPLE MEANS OUT THIS FAR HAVE VIRTUALLY NO CHANCE OF OCCURRING FOR THIS POPULATION. 32. {Test scores narrative} Suppose you randomly select a single individual from this population. Where would you expect his/her test score to fall? ANSWER: 95% OF THE TIME IT WOULD FALL BETWEEN 67 AND {Test scores narrative} Suppose you randomly select a sample of size 100 from this population. Where would you expect their average test score to fall? Compare your answer to what you would expect from a single individual selected at random from this population. ANSWER: WE EXPECT THE AVERAGE OF 100 TEST SCORES TO BE BETWEEN 76 AND 78. WE EXPECT AN INDIVIDUAL TO SCORE BETWEEN 67 AND {Test scores narrative} Find and compare the answers to the following two questions; explain why your answers are the same or different. 1) One individual is selected at random from the population. What range of test scores is reasonable to expect for this person? 2) A sample of 100 individuals is selected at random from the population. What range of average test scores is reasonable to expect for this group? ANSWER: WE EXPECT AN INDIVIDUAL SCORE TO BE BETWEEN 67 AND 87. WE EXPECT THE AVERAGE OF 100 TEST SCORES TO BE BETWEEN 76 AND 78. THE MEAN FOR A SAMPLE OF SIZE 100 HAS A SMALLER STANDARD DEVIATION AND GIVES A TIGHTER RANGE OF POSSIBLE VALUES. 35. {Test scores narrative} Suppose that the test scores are not bell-shaped, but are skewed to the right. You want to take a random sample and estimate the average test scored for this population. You want to be able to use the rule of sample means to interpret your results in this case. Under what (if any) conditions is this possible? ANSWER: YOU CAN USE IT ONLY IF THE SAMPLE IS LARGE ENOUGH (AT LEAST 30). 36. Explain, in words that a non-statistics student would understand, why the standard deviation of the various sample means taken from a population is smaller than the standard deviation of the individuals in the population? (In the first situation, assume all samples are of the same size, and that size is large.) ANSWER: SAMPLE MEANS VARY LESS, BECAUSE THEY ARE BASED ON MORE INFORMATION Which of the following are examples where you would be interested in estimating the population mean? a. About how long do left-handed people live? b. Do most people support or oppose the President s foreign policy? c. What size was the viewing population who tuned in to the ABC News special last night? d. All of the above.

6 38. In which of the following situations does the rule for sample means not apply? a. A pollster takes a random sample of 1,000 Americans and asks them to give their opinion of the President on a scale from 1 (completely disapprove) to 100 (completely approve). He is interested in the average rating. b. You take a random sample of 20 students scores from the ACT exam and record the average score. Assume ACT scores are bell-shaped. c. A sports fan takes a random sample of 20 NBA players and records their salaries. He wants to estimate the average salary for the entire NBA. d. The rule for sample means does not apply in any of these situations. ANSWER: C 39. If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from means from the various samples will have what approximate shape? a. A flat shape; each outcome should be equally likely. b. A bell-shape. c. A histogram. d. Unknown; it can change every time. ANSWER: B 40. Larger samples tend to result in accurate estimates of population values than do smaller samples. ANSWER: MORE 41. Samples of size 2,500 will produce estimates of the population value that are times more accurate than samples of size 25. (Assume the population is bell-shaped. Use standard deviation as a measure of accuracy.) ANSWER: 10