Introduction to Statistical Methods

Multiple Choice Questions (1 point each) 1. In order to determine if smoking causes cancer, researchers surveyed a large sample of adults. For each adult they recorded whether the person had smoked regularly at any period in their life and whether the person had cancer. They then compared the proportion of cancer cases in those who had smoked regularly at some time in their lives with the proportion of cases in those who had never smoked regularly at any point in their lives. What type of study is this? A) An observational study. B) An experiment, but not a double-blind experiment. C) A double-blind experiment. D) A block design. The explanatory variable (smoking) was not assigned to the subjects. They just simply recorded their answers about smoking. Thus, it’s an observational study. 2. Many studies are trying to find a cure for chronic back pain. In one such study, a physician is comparing the medication currently being used (drug A) to a newly developed drug (drug B). 73 volunteers, suffering from chronic back pain, are participating in this study. The physician’s assistant has a list of all 73 subjects and randomly divides the subjects into two groups. Group 1 will receive drug A and Group 2 will receive drug B. The assistant is the only person who knows to which group the subjects have been assigned. The physician monitors the subjects over a 2-month period and the amount of improvement is recorded. What type of study is this? A) An observational study. B) An experiment, but not a double-blind experiment. C) A double-blind experiment. D) A matched pairs experiment. The explanatory variable (drug A/B) was assigned randomly to the subjects. Thus, this is an experiment. And since the assistant is the only person who knows to which group the subject have been assigned, it is a double-blind experiment. 3. A study is designed to determine whether grades in a statistics course could be improved by offering special review material. The 250 students enrolled in a large introductory statistics class are also enrolled in one of 20 lab sections. The 20 lab sections are randomly divided into 2 groups of 10 lab sections each. The students in the first set of 10 lab sections are given extra review material during the last 15 minutes of each weekly lab session. The students in the remaining 10 lab sections receive the regular lesson material, without the extra review material. What type of study is this? A) An observational study. B) An experiment, but not a double-blind experiment. C) A double-blind experiment. D) A matched pairs experiment. The explanatory variable (extra review material given/not given) was randomly assigned to the students. Thus, it is an experiment. Since the students (and the lab assistants) knew if the they received extra review material, or did not receive it, it wasn’t a double-blind experiment. Use the following to answer questions 4–6: A marketing research firm wishes to determine if the adult men in Laramie, Wyoming, would be interested in a new upscale men’s clothing store. From a list of all residential addresses in Laramie, the firm selects a simple random sample of 100 and mails a brief questionnaire to each. 2 4. What is the population of interest? A) All adult men in Laramie, Wyoming. B) All residential addresses in Laramie, Wyoming. C) The members of the marketing firm that actually conducted the survey. D) The 100 addresses to which the survey was mailed. That’s what the marketing research firm wishes to determine: if the adult men in Laramie, Wyoming, would be interested in a new upscale men’s clothing store. 6. One particular neighborhood in Laramie happens to have exactly 100 residential addresses. What do we know about the chance that all 100 homes in that neighborhood end up being the sample that is selected? A) It is the same as for any other set of 100 residential addresses. B) It is exactly 0. Simple random samples will spread out the addresses selected. C) It is reasonably large due to the “cluster” effect. D) It is 100 divided by the size of the population of Laramie. E) It is much less likely than most random samples of 100 residential addresses are. This is the definition of simple random sample: every possible sample of size 100 has the same chance to be selected. No group of size 100 has lower or higher chance to be selected. 7. A large university wishes to determine the percentage of its students that have committed some form of academic dishonesty, such as cheating on an examination or plagiarism on assignments during their academic career. To determine this percentage, a random sample of their current students is selected. Each selected student is then interviewed by a staff member and asked if they had cheated. The results of this survey likely will be unreliable because A) the students likely will refuse to answer the question. B) those students who answer the question may not do so honestly. C) the interviewer being a staff member may be intimidating and hence there may be response bias. D) All of the above are reasons for concern. 8. A telemarketing firm in Los Angeles uses a device that dials residential telephone numbers in the city at random. Of the first 100 numbers dialed, 48% are unlisted. This is not surprising because 52% of all Los Angeles residential phones are unlisted. A) the 52% is a parameter, and the 48% is a statistic B) the 52% is a statistic, and the 52% is a parameter C) both the 52% and the 48% are parameters D) both the 52% and the 48% are statistics 48% is the sample statistic since it refers to the proportion of the sample (100 numbers dialed). 52% is the population parameter since it refers to the proportion of ALL Los Angeles residential phones. 9. A polling agency took a random sample of 1000 likely voters in Florida (population:about 18 million ), and another random sample of 1000 likely voters in New Mexico (population: about 2 million). A) the sampling variability of the Florida poll will be greater than that of the New Mexico poll B) the sampling variability of the Florida poll will be less than that of the New Mexico poll C) the sampling variability of the Florida poll will be about the same as for the New Mexico poll 3 Remember the soup pots? The size of the population does NOT matter regarding the sampling variability, only the sample size. Since the samples sizes were the same here, C is the correct choice. 10. True or False? (1 point each) T F When the z-score of a data value is 1.8, that means that the data value is 1.8 standard deviations below the mean. No, it means that the data value is 1.8 standard deviations ABOVE the mean. T F An appropriate notation for the fraction of all American adults who received at least one speeding ticket last year is p$ . No, since ALL American adults is our population, we use p for the proportion of the population. T F An appropriate notation for the mean distance traveled in a year by a sample of truck drivers is x . Yes. Since we are talking about a sample mean, the notation we use for that is x-bar. T F We cannot predict the likely accuracy of an estimate obtained from a sample if the sample is not taken randomly. Yes, the sample must be taken randomly. T F Usually a parameter value will fall within the interval specified by the point estimate plus and minus its margin of error, but this is not guaranteed to happen. Yes, we are usually pretty confident that the population parameter falls into the confidence interval, but it’s not guaranteed. 11. (5 points) Fill in the blanks. To make it easier for you, here’s the list of possible terms to use: Sampling variability Sampling design Sampling distribution Sampling error Sampling frame A list of potential individuals to be sampled is called ____sampling frame______ . Simple random sampling, stratified random sampling, and multistage random sampling are the most common types of ______sampling design____________ . The fact that different random samples from a population will give somewhat different results is called ____sampling variability_________. The distribution of the sample statistic for all the possible SRSs of the same size from the same population is called __sampling distribution______ . The fact that even well constructed samples will give results that are somewhat different from the population value simply because the entire population is not sampled is called __sampling error_ . 12. (14 points) Let’s suppose you are majoring in Child Development. As your project, you need to estimate the proportion of elementary school students in a small district who believe in the Easter Bunny. If you could ask every elementary student in the district, that would be called a(n) ___census_____. But you don’t have the time to ask each and every elementary kid about the Easter Bunny, so you take a random sample of 500 of them. 4 a. Clearly identify in words the population of interest, the parameter, the sample, and the statistic: Population: ALL elementary students in the district. Parameter: The proportion of ALL elementary students in the district who believes in the Easter Bunny. Sample: The 500 elementary students randomly selected from the districts. Statistic: The proportion of the elementary students in the SAMPLE who believes in the Easter Bunny. b. There are five elementary schools in the district, and each school has grades from Kindergarten to 5th grade. There are many ways you can pick those 500 students. Identify the sampling method described in each case below: From each elementary school you randomly select 100 students. Sampling method: ________stratified sample_______ You randomly pick two elementary schools, and then in these two schools you randomly select three grade levels. Then from these grades you randomly select 500 students. Sampling method: _______multistage sample_______________ From a list of all of the elementary school children in the district, you randomly select 500 students. Sampling method: _____simple random sample___________ c. True or False? T F If we would do the same survey with 500 students in a large school district with 20 elementary schools, the accuracy of the estimate of the proportion of all elementary kids who believe in Easter Bunny would not be as good as the one from the small district. No, again, remember the soup pots! The size of the population does NOT matter in the accuracy of the estimate, only the sample size. T F If you had a partner for your project, you could ask twice as many kids. Then the margin of error of our estimate of the proportion of all elementary kids in the district who believe in the Easter Bunny would be smaller. Yes, as the sample size increases, the standard deviation of the sampling distribution decreases. Thus, the margin of error decreases too. 13. (10 points) A pharmaceutical company wants to test the effectiveness of a new allergy drug. The experiment is described as a placebo-controlled, double-blind study. They have 300 subjects 5 available who suffer from severe allergies. After six months, the subjects’ symptoms are studied and compared. a. Identify the explanatory variable, and the response variable. Explanatory variable: drug taken (new allergy drug/placebo) Response variable: allergy symptoms b. Draw the outline of the experiment. Make sure you indicate the response variable, and group sizes. Group 1: 150 subjects New allergy drug Compare the subjects’ allergy symptoms after six months 300 subjects Group 2: 150 subjects Placebo c. The experimenters suspect that gender could be one of the lurking variables. Explain briefly what they could do to control for this variable. They could do a block design separating males and females first, then do the experiment separately with each gender. d. Explain where the experimenters should use random assignment: When they decide who goes to Group 1 and group 2. That should be randomly assigned to the subjects. 14. (6 points) According to the 2001 Nielsen ratings, Survivor II was one of the most-watched television shows in the United States during every week that it aired. Suppose that the true proportion of U.S. adults who watched Survivor II is p = 0.37. Indicate which one of the figures below shows the results of a simulation that drew 1000 SRS's of size n = 100 from a population with p = 0.37, and which one shows the results of another simulation that drew 1000 SRS's of size n = 1000 from a population with p = 0.37. n= 100 n = 1000 6 Explain how the histograms above demonstrate what the Central Limit Theorem says about the sampling distribution for a sample proportion. Discuss shape, center, and spread. Shape: the shape of the sampling distribution for a sample proportion should get closer and closer to the a bell-shaped curve as the sample size increases, and as the sample size increases more the distribution gets “skinnier” . Center: the centers of the distributions are the about the same, 0.37. The center of the sampling distribution is the population parameter, regardless of the sample size. Spread: the second distribution is “skinnier”, that is has less spread. And that’s how it should be. As the sample size increases the standard deviation of the distribution gets smaller and smaller. 15. (10 points) A USA Today poll asked a random sample of 1,012 U.S. adults who eat cereal what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 66% said that they drink it. Suppose that 70% of all U.S. adults who eat cereal actually drink the cereal milk. a. Fill in the blanks: p$ = __66% = 0.66___ p = ___70% = 0.70_____ b. If we would create the sampling distribution the sample proportions for all possible samples of size 1,012 for, what would be the mean and standard deviation of this sampling distribution? Mean: µ p$ = p = 0.70 Standard deviation: σ p$ = p(1 − p) = n 0.7(1 − 0.7) = 0.0144 1012 c. Are the conditions satisfied to assume that the shape of the sampling distribution is approximately normal? Check the conditions. np = 1012(0.7) =708.4 >10 n(1-p) = 1012(0.3) = 303.6 > 10 Yes, the conditions are satisfied to assume that the shape of the sampling distribution is approximately normal. d. About how many standard deviations is the survey percentage 66% from p? z= p$ − mean 0.66 − 0.7 = = −2.78 s. d . 0.0144 66% is about 2.8 standard deviations below the mean which is 70%. Based on this, do you have any doubts about the poll? Explain briefly. 7 Yes, I do have doubts. The observed sample statistic, 66%, is almost three standard deviations below the mean. That is an unusual value. If 70% of all U.S. adults who eat cereal actually drink the cereal milk, then observing a sample proportion of 66% or less would rarely happen. In fact, if you look up in the ztable, you see that the probability of this observation or less is only 0.27%. 16. (6 points) According to Harper's magazine, in the U.S. the time children spend watching television per year follows a normal distribution with mean of 1500 hours and a standard deviation of 250 hours. a. What percent of children watch television for less than 1200 hours per year? Draw the distribution, shade in the area that represents the percentage, and find its value. z= 750 b. 1000 1250 1200 1500 1750 2000 2250 x−µ σ = 1200 − 1500 = −12 . 250 From the z-table, or using the calculator, the percentage of children who watch television for less than 1200 hours per year is about 11.5%. Solve the appropriate equation and fill in the blank: 10% of all children in the U.S. watch more than _____ hours of television per year. 10% in the upper tail
90% in the lower tail. Thus, the corresponding z-score is 1.28. Then, x = z ⋅ σ + µ = 128 . (250) + 1500 = 1820 Thus, 10% of all children in the U.S. watch more than 1820 hours of television per year. 8 17. EXTRA CREDIT: The histograms below show four sampling distributions of statistics intended to estimate the same parameter. Label each distribution relative to the others as having large or small bias and as having large or small variability. (Circle the correct choices.) Bias: large Variability: large Bias: large Variability: large small small small small Bias: large Variability: large small small Bias: large Variability: large small small 9