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MATH 243 (PHILLIPS, SPRING 2009): SAMPLE MIDTERM PROBLEMS
Bring your student ID! The midterm exam will be mostly multiple choice or fill in the blank. In particular, many questions that ask for answers here will be multiple choice or fill in the blank on the midterm. A small number of problems on the real midterm will require work to be shown etc. Likely examples include experiment designs, confidence intervals, and hypothesis tests. Material from the lecture of Thursday 30 April will be in the exam. There are many more problems in this list than there will be on the midterm. In this list of sample problems, some continue across page breaks. Thus, you might find four answers to a multiple choice problem on the same page as the problem, but the correct answer is on the next page. General comments:
- Besides these problems, see all the material related to the homework, the quizzes, the examples in the book, etc.
- Many variations of these problems are possible. In particular, the questions could be asked in different ways, something which is multiple choice here could be short answer on the real exam, and the scatterplots and histograms could have very different features. For example, a problem might ask something about a stemplot, or for information about quartiles instead of the mean, etc. Thus, a problem could test the same ideas but superficially not look like anything here.
- There will be something on the exam which is not covered in the sample problems.
- The number of problems of a given type which appear here need not represent the weight given to a topic on the actual midterm. The choice of problem here is partly intended to reflect the possible variety of questions on the midterm.
- Recall that you may bring one 8. 5 × 11 page of notes and a calculator; see the syllabus for details.
- Recall that no calculator help will be provided on the exam. If you need to, write instruc- tions for your calculator on your 8. 5 × 11 page of notes.
- Recall that grading questions must be in writing and turned at the beginning of the lecture after the exams for your discussion section are returned. (This will be Tuesday 12 May.) Some specific comments on the problems:
- Always pay attention to whether the data were properly collected!
- In all problems asking for an experimental design, you must include information about the treatment groups and the response variable. Also, be sure that one can tell from your description that your experiment has all the characteristics expected of experiments of the type asked for. (It isn’t enough to just use words like “double blind”, “placebo”, etc.)
- A correct solution to a hypothesis test must state the conclusion in terms of the original language of the problem. Just saying that we reject the null hypothesis, or conclude that μ < μ 0 (or something similar), or, if appropriate, that we do not reject the null hypothesis, etc., will not get full credit.
- Solutions to confidence interval problems must show the critical value you used (z∗).
- Explain clearly what it means for a point in a scatterplot to be influential for the regression line.
Date: 30 April 2009. 1
(Not all the steps have the same point value.) Lobotozine is a new recreational drug which causes severe mental impairment even in low doses. You have bought an apparatus which measures lobotozine concentrations in blood specimens in the range 10–500 micrograms/liter (μg/l). The results for any given specimen are approximately normally distributed with standard deviation σ = 2 μg/l. The mean is supposed to be the true concentration, but you suspect the manufacturer calibrated it incorrectly, so that the mean differs from the true concentration. You test this apparatus by using it to make 5 measurements on a reference specimen in which the concentration of lobotozine is known to be 100 μg/l. The results are:
6 103. 6 102. 1 100. 6 103. 1
Assuming that the appropriate statistical procedure is safe to use, is this convincing evidence that the calibration is incorrect, at significance level α = 0.05? Answer using the following steps.
(1) State the hypotheses you will test. (2) State the test you will use. (3) Calculate the test statistic. (4) Give a P -value (or give two values between which P lies). (5) Draw the appropriate conclusion, expressing it in words appropriate for the context of the problem.
- (Not all the steps have the same point value.) Math SAT scores vary normally with mean 516 and standard deviation 114. You suspect that high school seniors in Mudville have, on average, lower Math SAT scores than the general population of people who take the Math SAT. You choose a simple random sample of 8 high school seniors in Mudville who took the Math SAT. Their Math SAT scores are:
550 590 430 390 490 270 300 380.
Assume that the standard deviation of Math SAT scores of high school seniors in Mudville is the same as for the general population. Is this convincing evidence that high school seniors in Mudville who take the Math SAT do worse than the general population of test takers, at significance level α = 0.02? Answer using the following steps.
(1) State the hypotheses you will test. (2) State the test you will use. (3) Calculate the test statistic. (4) Give a P -value (or give two values between which P lies). (5) Draw the appropriate conclusion, expressing it in words appropriate for the context of the problem.
3a. A sociologist believes that high school seniors in Hicksville on average do better on the Math SAT than the general population of high school seniors. He tests this belief at significance α = 0.05, using the one sample z procedure, by choosing a simple random sample of 50 high school seniors in Hicksville. His test gives a P -value of 0.0772. What is the correct conclusion of the sociologist’s test?
a. He does not reject the null hypothesis, and concludes that there is strong evidence that high school seniors in Hicksville do not do better on the Math SAT than the general population of high school seniors.
f. Ha: μ < 176. g. No hypotheses are appropriate, because of improper data collection. h. None of the above.
- A test of significance does which of the following:
a. Determine whether the sample or experiment was properly designed. b. Decide whether one can reasonably rule out the observed effect being due to chance. c. Decide whether one can reasonably rule out the observed effect being unimportant. d. Determine how large the observed effect is. e. Determine whether the experimenter is knowledgeable about statistics. f. Decide whether the problem being investigated is important. g. All of the above.
- I want to decide whether the mean GPA of students in my Math 243 class is larger than the mean GPA of all University of Oregon students. I obtain from the registrar the mean μ 0 and standard deviation σ of the GPAs of all University of Oregon students. Then I find the mean GPA of the 10 students in my class who sit in the front row. Using the one sample z statistic, I find that I can reject the null hypothesis (that the mean GPAs are the same) at significance level α = 0.05. Which of the following is it safe to conclude?
a. There is strong evidence that the mean GPA of students in my Math 243 class is larger than the mean GPA of all University of Oregon students. b. While the difference may be small, a difference as or more extreme than that observed in the group is unlikely to have arisen by chance if the mean GPA of students in my Math 243 class is the same as the mean GPA of all University of Oregon students. c. The mean GPA of students in my Math 243 class is probably much larger than the mean GPA of all University of Oregon students. d. Nothing, because of a poor choice of sampling method. e. The mean GPA of students in my Math 243 class is larger by at least 0.05 than the mean GPA of all University of Oregon students. f. Students who sit in the front row have higher GPAs than those who sit elsewhere in my class.
- A local business consultant wants to know whether Eugene residents are planning to spend more on holiday shopping this year than last. Previous research has shown that Eugene residents spent an average of $719 on holiday shopping last year. The consultant interviews a simple random sample of 70 Eugene residents, asking them how much they are planning to spend this year. He performs a one sample z test on the results, and rejects the null hypothesis (that Eugene residents plan to spend the same amount this year as they spent last year) at the significance level α = 0. 01. Which of the following is it safe to conclude?
a. There is strong evidence that Eugene residents plan to spend more than $719 this year. b. While the difference may be small, a difference as or more extreme than that observed in the group is unlikely to have arisen by chance if Eugene residents plan to spend $719 this year. c. Eugene residents plan to spend much more than $719 this year. d. Nothing, because of a poor choice of sampling method. e. Eugene residents plan to spend at least 1% more than $719 this year.
f. People who shop at the Gateway Mall probably plan to spend more than other Eugene residents do.
- I am testing a two sided alternative hypothesis on a population for which the standard deviation is known. If the z test statistic is z = 1.53, then the P -value for this test is:
a. 0. 9370 b. 0. 0630 c. 0. 1260 d. 1. 8740 e. Impossible to tell from the given data. f. None of the above.
- Choose the answer which, when put in the blank, makes a true sentence. The P -value is a. computed assuming that H 0 is false. b. computed assuming that H 0 is true. c. the probability that H 0 is false. d. the probability that H 0 is true. e. None of the above.
- The best description of the term statistically significant is:
a. An observed effect which satisfies the criteria of the “1. 5 × IQR” rule. b. An observed effect which could never occur by chance. c. An observed effect so large that it would rarely occur by chance. d. An observed effect that cannot be explained. e. None of the above.
- Consider the following histogram (made from integer data):
The distribution of the data plotted is:
a. Roughly symmetric with no outliers. b. Roughly symmetric with one or more outliers. c. Skewed to the right with no outliers. d.
- A relatively small set of data (between 10 and 20 items) is roughly normally distributed, except for one extreme outlier at the low end. The statistician analyzing these data has good reason to believe this outlier resulted from a measurement error, and therefore discards it. How does the new mean (of the data without this outlier) compare with the mean of the original data?
a. The new mean is significantly smaller than the old mean. b. The new mean is about the same as the old mean. c. The new mean must be exactly the same as the old mean. d. The new mean is significantly larger than the old mean. e. Any of the above can happen. f. None of the above.
- A relatively small set of data (between 10 and 20 items) is roughly normally distributed, except for one extreme outlier at the low end. The statistician analyzing these data has good reason to believe this outlier resulted from a measurement error, and therefore discards it. How does the new median (of the data without this outlier) compare with the median of the original data?
a. The new median is significantly smaller than the old median. b. The new median is about the same as the old median. c. The new median must be exactly the same as the old median. d. The new median is significantly larger than the old median. e. Any of the above can happen. f. None of the above.
- A relatively small set of data (between 10 and 20 items) is roughly normally distributed, except for one extreme outlier at the low end. The statistician analyzing these data has good reason to believe this outlier resulted from a measurement error, and therefore discards it. How does the new first quartile (of the data without this outlier) compare with the first quartile of the original data?
a. The new first quartile is significantly smaller than the old first quartile. b. The new first quartile is about the same as the old first quartile. c. The new first quartile must be exactly the same as the old first quartile. d. The new first quartile is significantly larger than the old first quartile. e. Any of the above can happen. f. None of the above.
- A relatively small set of data (between 10 and 20 items) is roughly normally distributed, except for one extreme outlier at the low end. The statistician analyzing these data has good reason to believe this outlier resulted from a measurement error, and therefore discards it. How does the new third quartile (of the data without this outlier) compare with the third quartile of the original data?
a. The new third quartile is significantly smaller than the old third quartile. b. The new third quartile must be exactly the same as the old third quartile. c. The new third quartile is about the same as the old third quartile. d. The new third quartile is significantly larger than the old third quartile. e. Any of the above can happen. f. None of the above.
- A relatively small set of data (between 10 and 20 items) is roughly normally distributed, except for one extreme outlier at the low end. The statistician analyzing these data has good reason to believe this outlier resulted from a measurement error, and therefore discards it. How does the
new standard deviation (of the data without this outlier) compare with the standard deviation of the original data?
a. The new standard deviation is significantly smaller than the old standard deviation. b. The new standard deviation is about the same as the old standard deviation. c. The new standard deviation must be exactly the same as the old standard deviation. d. The new standard deviation is significantly larger than the old standard deviation. e. Any of the above can happen. f. None of the above.
- A density curve is shown below with three points marked. Identify the mean and median by filling in the blanks with the appropriate letter.
a (^) b c
Mean
Median
- Ken Griffey Jr’s home run totals for each season from 1989 to 2008 are listed
16 22 22 27 45 40 17 49 56 56 48 40 22 8 13 20 35 27 30 18 (1) Is 56 an outlier?
(2) Find the mean number of home runs Griffey hit.
(3) Find the standard deviation of Griffey’s home run totals for his first four seasons. (Do this by hand.)
(4) Look at the mean, median and histogram. Which of the following describes the distribu- tion? a. The distribution is skewed left and has no outliers. b. The distribution is skewed right and has no outliers.
- Wang’s Widgets Inc. has ten employees. The mean salary is $52, 100 , the standard deviation is $24,049 (to the nearest dollar), and the five number summary is
$25, 800 $37, 000 $43, 200 $61, 000 $102, 000.
The owner has decided to give the highest paid person a $200,000 raise. (He did something that earned the company millions in extra profits.) In all parts of this problem, include appropriate units. a. What is the new mean salary? Why?
b. What is the new median salary? Why?
c. Suppose that instead the owner gives everyone a $10,000 raise. What is the new standard deviation? Why?
- The masses (in kilograms) of crumple-horned snorkacks are normally distributed with mean 70 and standard deviation 8. The masses (in kilograms) of spiral-horned snorkacks are normally distributed with mean 66 and standard deviation 4. The masses (in kilograms) of purple-spotted snorkacks are normally distributed with mean 45 and standard deviation 5. The Zoo of Magical Creatures has a 56 kilogram crumple-horned snorkack, a 58 kilogram spiral-horned snorkack, and a 50 kilogram purple-spotted snorkack. Which of these is the most underweight for its species?
a. The crumple-horned snorkack. b. The spiral-horned snorkack. c. The purple-spotted snorkack. d. The crumple-horned snorkack and the spiral-horned snorkack are equally underweight. e. Cannot be determined from the information given. f. None of the above.
- The heights of fourth graders in Seattle are approximately normally distributed with mean 46 .5 inches and standard deviation 4.8 inches.
a. Approximately what percentage of fourth graders is taller than 41.7 inches?
b. Approximately 40% of the fourth graders are shorter than inches.
c. Approximately what percentage of simple random samples of size 16 of fourth graders has sample mean more than 48.9 inches?
- The weights of zucchinis harvested in a certain California valley are approximately normally distributed with mean 2.41 pounds and standard deviation 0.62 pounds. A truckload of zucchinis is taken to market. In each of the following parts, show your work, and draw a picture of the appropriate normal curve with the relevant points on the horizontal axis clearly marked and with the appropriate area shaded and clearly identified. The curve must look reasonable.
a. Approximately what percentage of the zucchinis weigh more than 1.17 pounds?
b. Approximately what percentage of the zucchinis weigh between 1.79 and 3.65 pounds?
c. Approximately 10% of the zucchinis weigh more than pounds.
- In 2002, Jane Smith got a score of 690 on the mathematics part of the SAT, and John Doe got a score of 28 on the ACT Assessment mathematics test. SAT math scores in 2002 were normally distributed with mean 516 and standard deviation 114, and ACT math scores in 2002 were normally distributed with mean 20.6 and standard deviation 5.0. Find the standardized scores for both students. Assuming both tests measure the same kind of ability, who did better?
- Two variables in a study are said to be confounded if what?
a. One cannot distinguish their effects on a response variable. b. They are highly correlated. c. They do not have a normal distribution. d. One of them is a placebo. e. Both are lurking variables. f. The statistician conducting the study is incompetent.
- Consider the following scatterplot:
5 10 15 20 25
5
10
15
20
25
The association of the variables plotted is:
a. Clearly positive and roughly linear. b. Clearly positive and clearly nonlinear. c. Neither clearly positive nor clearly negative. d. Clearly negative and roughly linear. e. Clearly negative and clearly nonlinear.
- Consider the following scatterplot:
- Consider the following scatterplot:
Give the approximate coordinates of all outliers. If there are none, write “NONE”.
- John Doe’s score on Midterm 1 was 96, but he missed Midterm 2 due to illness. You are to predict what his score would have been, given the following information on the scores of all students who took both Midterm 1 and Midterm 2.
Midterm 1: Five number summary 22 51 77 82 99; mean 69; standard deviation 23. 63. Midterm 2: Five number summary 15 42 71 84 98; mean 65; standard deviation 26. 80. Correlation r ≈ 0 .8570; r^2 ≈ 0. 7345. (All noninteger values given to 4 significant digits.)
- A scientist studying the fire-breathing monsters of the planet Yuggxth wants to know if egg size is a good predictor of adult mass for these creatures. At considerable personal risk, he has managed to obtain the following data on five individuals.
Individual Egg diameter (cm) Adult mass (tons) A 20 19 B 10 12 C 15 18 D 8 14 E 15 14
a. Draw a scatterplot on the axes provided. Be sure to label your axes, and make an appro- priate choice of which variable to put on the horizontal axis. [Grid omitted in sample problem.]
b. Find the equation of the least squares regression line and plot it on the graph above. c. What is the correlation between egg diameter and adult mass? What percentage of the variation in adult mass is explained by egg diameter?
- Consider the following scatterplot:
10 20 30 40 50
5
10
15
20
25
Identify all influential points for the regression line by giving their approximate coordinates.
- As part of an investigation of the mean Math SAT score of University of Oregon students, the Registrar’s office selects 100 students from the University of Oregon, and looks up their Math SAT scores in their application materials. What is the sample?
a. The Registrar’s office. b. All University of Oregon students who took the Math SAT. c. The 100 selected students. d. The Math SAT scores of the 100 selected students. e. The Math SAT scores of all University of Oregon students who took the Math SAT. f. Impossible to tell from the information given. g. None of the above.
The next two problems ask the same question about slightly different situations.
- A political scientist wants to know how University of Oregon students feel about the Social Security system. He chooses a simple random sample of 398 students and sends a survey to their university assigned email addresses. Only 87 surveys are returned. What is the sample in this study?
a. All college students. b. All University of Oregon students. c. The 398 students who were emailed the survey. d. The 87 students who returned the survey. e. The 211 students who did not return the survey. f. None of the above.
- A national news organization conducted a telephone survey to determine what portion of registered voters support a particular presidential candidate. The reported margin of error was ±3%. Which of the following common sources of error are not accounted for in the reported margin of error?
a. The fact that the survey omitted Alaska and Hawaii. b. The fact that people without telephones were missed.
- A class has 12 students: Nathan, Zach, Kevin, Gary, Bob, Miles, Mei-Chu, Janet, Tom, Louise, Christie, Steve. Start at line 131 in Table B to select a simple random sample of 3 students.
- The Nurses’s Health Study has interviewed a sample of more then 100,000 female registered nurses every two years since 1976. The study finds that “light to moderate drinkers had a signifi- cantly lower risk of death” than either nondrinkers or heavy drinkers. The Nurse’s Health Study is an example of:
a. An observational study. b. An experiment. c. A double blind design. d. This has elements of an observational study and an experiment. e. None of the above.
- Many studies have found that people who drink alcohol in moderation have lower risk of heart attacks than either nondrinkers or heavy drinkers. Does alcohol consumption also improve survival after a heart attack? One study followed 1913 people who were hospitalized after severe heart attacks. In the year before their heart attack, 47% of those people did not drink, 36% drank moderately, and 17% drank heavily. After four years, fewer of the moderate drinkers had died. What is the response variable in this study?
a. The amount of alcohol the people drank. b. The rate of severe heart attacks. c. The length of time hospitalized after a severe heart attack. d. Survival or nonsurvival after four years. e. The survival rate after a severe heart attack. f. Alcohol consumption. g. None of the above.
- The National Halothane Study was a major investigation of the safety of anesthetics used in surgery. Records of over 850,000 operations performed in over 34 major hospitals. Anesthetics A, B, and D all had death rates under 2%, while anesthetic C had a death rate of 3.4%. We can conclude that:
a. Anesthetic C is causing more deaths than the other three. b. The sample size is not sufficient to conclude anything definitive. c. Due to the possible presence of lurking variables confounding the results, we cannot con- clude anything definitive. d. Due to nonresponse issues, we cannot conclude anything definitive. e. None of the above.
- Which of the following statements about a table of random digits is true?
a. In a line of 40 digits, there are exactly four 0’s. b. In a line of 40 digits there can never be a easily seen pattern, such as the same four digits repeating.
c. Every pair of digits has a 1/100 chance of being 00. d. Every digit must appear at least once in the table. e. None of the above.
- A study of cell phones and the risk of brain cancer looked at a group of 469 people who have brain cancer. The investigators matched each cancer patient with a person of the same sex, age, and race who did not have brain cancer, then asked about use of cell phones. This is:
a. A controlled experiment. b. A matched pairs design. c. An uncontrolled experiment. d. A double blind design. e. An observational study. f. None of the above.
- Can changing diet reduce high blood pressure? Vegetarian diets and low-salt diets are both promising. Men with high blood pressure are assigned at random to four diets: (1) normal diet with unrestricted salt; (2) vegetarian with unrestricted salt; (3) normal with restricted salt; and (4) vegetarian with restricted salt. This experiment has:
a. One factor, a choice of diet. b. One factor, the blood pressure of men on the diet. c. Two factors, normal vs. vegetarian diet and unrestricted vs. restricted diet. d. Four factors, the four diets being compared. e. None of the above.
- Research doctors test drugs by prescribing different amounts and observing the results on their patients. One question we could ask here is: “Does the amount of drug prescribed determine the length of the recovery time?” The explanatory variable is:
a. The drug being tested. b. The amount of drug prescribed. c. The length of the recovery time. d. The results on the patients. e. The disease the patients had. f. None of the above.
- An experiment to test whether gingko extract improves memory and concentration used as subjects 230 healthy people over 60 year old. These subjects were randomly assigned to two groups. Those in one group received pills containing gingko extract, while those in the other received placebo pills which looked and tasted the same. All subjects took a battery of tests for learning and memory before the treatment started, and again after six weeks. Neither the subjects nor the experimenters knew which subject got which treatment until after the experiment was complete. The experiment found no evidence that gingko extract improves memory and concentration. The purpose of randomization in this experiment was:
a. To ensure that each treatment group had the same number of men and women. b. To make all possible outcomes of the experiment equally likely.
- A gambler tosses a fair coin 7 times, and each time it comes up tails. What is the probability that the 8th toss will also be tails?
a. 0 b. 7/ 2 c. 1 d. 6/ 7 e. 1/ 2 f. 1/ 7 g. Impossible to determine from the information provided. h. None of the above.
- You roll two fair six sided dice. Consider the following events:
X: The sum of the rolls is greater than 9. Y : The product of the rolls is 25. Z: Both dice rolled odd numbers.
Which of the following statements about the probabilities of these events are true?
a. P (X or Y ) = P (X) + P (Y ). b. P (X or Z) = P (X) + P (Z). c. P (Y or Z) = P (Y ) + P (Z). d. P (X or Y or Z) = 1. e. None of the above.
- (3 points/part; total 6 points.) If you draw an M&M candy at random from a particular bag of M&M candies, the candy you draw will have one of six colors. The table below gives the probability that a randomly chosen M&M has each color.
Color Brown Yellow Red Green Orange Blue Probability 0.2 0.3 0.1 0.1? 0. Remember to show your work (even if you can do the problem in your head). a. Find the probability of drawing an orange M&M. b. Find the probability of drawing a yellow or green M&M.
- MC3. We choose a sample of 50 graduating seniors at the University of Oregon, ask for their grade point averages, and find that the mean of these 50 numbers is 2. 47. Meanwhile, the registrar tells us that the mean of the grade point averages of all graduating seniors is 3. 11. The number 2 .47 is what:
a. A parameter. b. A population. c. A statistic. d. A sample. e. Both (a) and (b). f. Both (c) and (d). g. None of the above.
- A simple random sample of size n is drawn from a population with mean μ and finite standard deviation σ. The Central Limit Theorem says that when n is sufficiently large:
a. The distribution of the population is exactly normal. b. The distribution of the population is approximately normal. c. The distribution of the sample mean is approximately normal. d. The distribution of the sample mean is exactly normal. e. The limit of the sample size is the center of the population size. f. The center of the population distribution is limited.
- As part of an investigation of the mean Math SAT score of University of Oregon students, the Registrar’s office selects 100 students from the University of Oregon, and looks up their Math SAT scores in their application materials. What is the parameter?
a. The Registrar’s office. b. All University of Oregon students. c. The 100 selected students. d. The mean of the Math SAT scores of the 100 selected students. e. The mean of the Math SAT scores of all University of Oregon students. f. Impossible to tell from the information given. g. None of the above.
- The diameters of the balls in ball bearings from Wang’s Bearings Inc. vary according to a normal distribution with mean 2.5 mm and standard deviation 0.1 mm. What is the sampling distribution for a simple random sample of 25 balls?
a. N (2. 5 , 0 .5). b. N (2. 5 , 0 .1). c. N (2. 5 , 0 .02). d. N (2. 5 , 0 .004). e. N (0. 5 , 0 .1). f. N (0. 5 , 0 .02). g. Impossible to determine from the information given.
- IQ test scores are normally distributed with mean 100 and standard deviation 11. 4. What is the probability that a randomly selected test-taker receives an IQ score between 120 and 125?
- SAT scores are normally distributed with mean 1030 and standard deviation 209. What is the probability that the mean score of 30 randomly selected test-takers is between 950 and 1000?
- A 95% confidence interval for a certain population mean has been calculated from a simple random sample of size 100. Assuming that the population standard deviation is known, which of the following will decrease the margin of error for the confidence interval?
a. Increasing the confidence level to 98%. b. Decreasing the sample size to 50. c. Increasing the sample size to 200.
