Learning Objectives:
Thinking about chance
• Explain how random events behave in the short run and in the long run and how random and haphazard are not the same thing.
• Perform basic probability calculations using die rolls and coin tosses.
• Define probability, and apply the rules for probability.
• Explain whether the law of averages is true.
• Explain how personal probability differs from a scientific or experimental probability.
Probability models
• Define a probability model. Create a probability model for a particular story's events.
• Apply the basic rules of probability to a story problem.
• Calculate probabilities using a probability model, including summing up probabilities or subtracting probabilities from the total.
• Define a sampling distribution.
The house edge: expected values
• Define expected value, and calculate the expected value when given a probability model.
• Define the law of large numbers, and explain how it is different from the mythical "law of averages."
• Explain how casinos and insurance companies stay in business and make money.
The Normal distribution
• Identify data that is Normally distributed.
• Discuss how the shape/position of the Normal curve changes when the standard deviation increases/decreases or when the mean increases/decreases.
• Define the standardized value or Z-score. Calculate the Z-score, and use the Z-score to do comparisons.
• Calculate probabilities and cut-off values using the 68%-95%-99.7% (Empirical) Rule.
• Identify the mean, standard deviation, cut-off value, probability, and Z-score on a Normal curve.
• Use the Normal table to get percentiles (probabilities) for forward problems and to get Z-scores
in order to determine cut-offs for backward problems using both > and < in the inequalities.
• Recognize whether a story is a forward or backward Normal distribution problem, and perform the appropriate calculations showing correct notation, the initial probability expression, and all necessary steps.
What is a confidence interval?
• Define statistical inference and explain when statistical inference is used.
• Explain what the confidence interval means and whether the results refer to the population or the sample.
• Calculate the margin of error and identify the margin of error in a confidence statement. Explain what type of error is covered in the margin of error.
• Determine whether a story is better described with a proportion or a mean.
• Use appropriate notation for proportions and means, both in the population and the sample.
• Calculate a confidence interval for a proportion and for a mean.
• Describe how increasing/decreasing the sample size or confidence level changes the margin of error (width of the confidence interval).
• Apply cautions for using confidence intervals.
o Need a simple random sample.
o Data must be collected correctly.
o No outliers.
Practice Exam Questions
These practice exam questions are not meant to be an exhaustive list. These are simply examples of questions that have been asked in the past. You definitely should try working through these examples with your cheat sheet, but make sure that you use the Learning Objectives to create your cheat sheet and to help you find other problems to practice. Your instructor uses the Learning Objectives when writing the exams. The odd-numbered problems in your book have answers in the back, and the StatsPortal website has many great resources to help your studying, especially the Learning Curve quizzes and the video series. Remember that you are allowed to bring a 1-page (both sides ok), handwritten-in-your-own-handwriting, 8 'A" x 11" cheat sheet with you to the exam. Your name should be on the cheat sheet, and it will be worth 1 point on the exam when you turn it in.
The experiment/sampling design and ethics topics from Exam 1 are important throughout the whole semester and may show up on later exams.
1. Before the 2008 presidential campaign, the Gallup Poll asked a sample of 1,000 people for whom they would vote for president; 52% said Obama. The margin of error for a 95% confidence interval announced by news reports of this poll was:
A) ±8% B) ±6% C) ±4% D) ±3% E) ±2%
Show your work:
2. "Margin of error" in this situation means that if there is no bias
A) every sample the Gallup Poll takes will come at least this close to the truth.
B) about half of the samples the Gallup Poll takes will come at least this close to the truth.
C) about 95% of all samples the Gallup Poll takes will come at least this close to the truth.
D) the sampling method is biased-otherwise the poll would always give the correct answer.
E) there are serious nonsampling errors-otherwise the poll would always give the correct answer.
3. A Gallup poll surveyed 3,112 voters. An AP poll surveyed 778 voters from the same population on the exact same question. How does the margin of error for a 95% confidence interval compare for the Gallup and AP polls?
a. The Gallup margin of error is bigger than the AP margin of error.
b. The Gallup margin of error is the same as the AP margin of error.
c. The Gallup margin of error is smaller than the AP margin of error.
4. The mean is 80 and the standard deviation is 10. What is the standard score for an observation of 90? Show your work below.
Suppose that the BAC of students who drink five beers varies from student to student according to a Normal distribution with mean 0.07 and standard deviation 0.01. For questions 10 through 13, show your work below the question and write your answer in the line.
5. The middle 99.7% of students who drink five beers have BAC between what two numbers?
6. What percent of students who drink five beers have BAC below 0.09?
7. What BAC do the highest 15% of students have after drinking five beers?
8. Sketch a Normal curve for the problem above with mean = 0.07 and standard deviation = 0.01. Label your x-axis from 0.04 to 0.10 in 0.01 increments. Also show the how the top 15% of students (from #7) would be represented on that Normal curve.
9. A poker player is dealt poor hands for several hours. He decides to bet heavily on the last hand of the evening on the grounds that after many bad hands he is due for a winner.
a) He's right, because the winnings have to average out.
b) He's wrong, because successive deals are independent of each other.
c) He's right, because successive deals are independent of each other.
d) He's wrong, because his expected winnings are $0 and he's below that now.
10. The probability of an outcome of a random phenomenon is
a) either 0 or 1, depending on whether the phenomenon can actually occur.
b) the proportion of a very long series of repetitions on which the outcome occurs.
c) the mean plus or minus two standard deviations.
d) the confidence level.
In government data, a family consists of two or more persons who live together and are related by blood or marriage. Choose an American family at random and count the number of people it contains. Here is the assignment of probabilities for your outcome:
# of persons 2 3 4 5 6 7 or more
Probability 0.42 0.23 0.21 0.09 0.03 ???
11. What is the probability a family will have 7 or more people in it? Show your work below.
12. What is the probability a family will have more than 2 people in it? Show your work below.
13. What is the probability a family will have 3 or 4 people in it? Show your work below.
14. What is the expected value of the number of family members? Show your work below and write your answer in the blank.
15. A deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise you lose $1. If you make this wager very many times, what will be the mean outcome?
a) about -$1, because you will lose most of the time.
b) about $9, because you will win $10 but lose only $1.
c) about -$0.15, that is, on the average you lose about 15 cents.
d) about $0.77, that is, on the average you win about 77 cents.
e) about $0 because the random draw gives you a fair bet.
16. A deck of 52 cards contains 13 hearts. Here is another wager: draw one card at random from the deck. If the card drawn is a heart, you win $2. Otherwise, you lose $1. Compare this wager (call it Wager 2) with that of the previous question (call it Wager 1). Which one should you prefer?
a) Wager 1, because it has a higher expected value.
b) Wager 2, because it has a higher expected value.
c) Wager 1, because it has a higher probability of winning.
d) Wager 2, because it has a higher probability of winning.
e) Both wagers are equally favorable.
17. A psychologist thinks that listening to Bach may help people think. She gives subjects a set of puzzles and measures how many they solve in 5 minutes while listening to Bach. From data on many people, the psychologist determines a probability model for solving 1, 2, 3,4 , and 5 puzzles solved. The expected value she calculates from this probability model is 2.6. The law of large numbers says
a) observe whether each of many subjects solves a puzzle. The proportion who solve a puzzle will be close to the expected value.
b) if you observe five subjects in a row who solve only one puzzle, the next several subjects are likely to solve three or four puzzles because the average must stay close to the expected value.
c) the expected value is correct only in a randomized comparative experiment.
d) observe many subjects and record how many puzzles each solves. The average will be close to the expected value.
The distribution of heights of adult men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Show your work. Answers without correct work will not receive any credit.
18. What percent of all men are shorter than 64 inches?
19. How tall is a man whose standardized height is z = -0.3? Answer in inches.
20. What percent of all men are taller than a man whose height is at the 60th percentile?
21. How tall is a man who is in the 82nd percentile? Answer in inches.
The casino game craps is based on rolling two dice. Here is the assignment of probabilities to the sum of the numbers on the up faces when two dice are rolled:
Outcome 2 3 4 5 6 7 8 9 10 11 12
Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
The most common bet in craps is the "pass line." A pass line bettor wins immediately if either a 7 or an 11 comes up on the first roll. This is called a "natural." Use this information to answer questions 11 through 14.
22. What is the probability of a natural?
a. 2/36 b. 6/36 c. 8/36 d. 12/36 e. 20/36
23. What is the probability you do not roll a 7?
a. 6/36 b. 28/36 c. 0 d. 30/36 e. 8/36
24. Gigi has rolled a natural on four straight tosses of the dice. This excites the gamblers standing around the table. They should know that:
a. Gigi has a hot hand, so she is more likely to roll another natural.
b. The law of averages says that Gigi is now less likely to roll another natural.
c. Rolls are independent, so the chance of rolling another natural has not changed.
d. Four straight naturals are almost impossible, so the dice are probably loaded.
e. They should not be surprised because the probability of four straight naturals is 2/36.
25. The table above shows a legitimate probability model because:
a. All the probabilities are between 0 and 1.
b. All the probabilities are between -1 and 1.
c. The sum of all the probabilities is exactly 1.
d. Both A and C.
e. Both B and C.
26. A game involving a pair of dice pays you $4 with probability 16/36, costs you $2 with probability 14/36, and costs you $6 with probability 6/36. What is the expected value of the amount of money you win or lose after one play of the game? Show your work using 3 decimal places.
An answer without correct work will receive no credit.
Fewer US teens smoke, drink than European peers: study
Fewer teenagers in the United States smoke and drink compared to their European counterparts, but more use drugs, according to a University of Michigan study released Friday.
Using data from 36 European countries plus the United States, researchers found that 27 percent of US adolescents had consumed alcohol in the month prior to being quizzed by pollsters, compared to 57 percent of Europeans.
Twelve percent of American teens had smoked tobacco, compared to 20 percent for the Europeans, according to the study, the fifth of its kind since 1995 with a total of 100,000 students aged 15 and 16 taking part.
"One of the reasons that smoking and drinking rates among adolescents are so much lower here than in Europe is that both behaviors have been declining and have reached historically low levels in the United States," lead author Lloyd Johnston said.
"But even in the earlier years of the European surveys, drinking and smoking by American adolescents was quite low by comparison," he said, adding however that "use of illicit drugs is quite a different matter."
Eighteen percent of the Americans had used marijuana or hashish, a proportion exceeded in Europe only in France (24 percent) and Monaco (21 percent).
On average, only seven percent of young Europeans had used either substance.
Relatively easy access to marijuana and little awareness of its dangers explain the figures, according to the responses that researchers collected from survey participants.
The Americans were also the biggest users of all other drugs besides marijuana -- such as LSD, ecstasy and amphetamines -- at 16 percent, compared to six percent across Europe.
"Clearly the United States has attained relatively low rates of use for cigarettes and alcohol, though not as low as we would like," Johnston said. "But the level of illicit drug use by adolescents is still exceptional here."
15,400 teenagers in the United States took part in the survey, along with at least 2,400 counterparts in each of the 36 European nations, the University of Michigan said in a statement.
27. What is the population of interest for this study?
28. Name the most likely type of error that could impact the findings in this study. Explain your choice.
29. What was the sample size for individuals from the US for the current year?
30. Calculate a 90% confidence interval (CI) for the proportion of U.S. teenagers that had used marijuana or hashish (again for the current year).
31. Suppose you were to change the confidence level in question 4 to 95% using the same sample. How would the confidence interval change? No calculations necessary.
a. The confidence interval would be the same width but shifted to the left.
b. The confidence interval would be the same width but shifted to the right.
c. The confidence interval would have the same center but would be wider.
d. The confidence interval would have the same center but would be narrower.
32. What if the sample size was only 5000 people for the 90% confidence interval in #4. How would the confidence interval change with this smaller sample size? No calculations necessary.
a. The confidence interval would be the same width but shifted to the left.
b. The confidence interval would be the same width but shifted to the right.
c. The confidence interval would have the same center but would be wider.
d. The confidence interval would have the same center but would be narrower.
33. A recent poll reported a confidence 95% confidence interval of 52% ± 3%. The poll was carried out by telephone, so people without phones are always excluded from the sample. Any errors in the final results due to excluding people without phones:
a. are included in the announced margin of error.
b. are in addition to the announced margin of error.
c. can be ignored, because these people are not part of the population.
d. can be ignored because this is a nonsampling error.
34. The phrase "95% confidence" means
a. our results are true for 9% of the population of all adults.
b. 95% of the population falls within the margin of error we announce.
c. the probability is 0.95 that a randomly chosen adult falls in the margin of error we announce.
d. we got these results using a method that gives correct answers in 95% of all samples.
35. Gallup polled 1,523 adults and 501 teens on whether they generally approved of legal gambling. 63% of adults and 52% of teens said yes. The margin of error for a 95% confidence statement about teens would be
a. greater than for adults, because the teen sample is smaller.
b. less than for adults, because the teen sample is smaller.
c. less than for adults, because there are fewer teens in the population.
d. the same as for adults, because they both come from the same sample survey.
e. Can't say, because it depends on what percent of each population was in the sample.
36. Which of the following methods would decrease the width of a confidence interval for a mean, if all else stays the same. You may choose more than one answer for this question.
a. Increase the level of confidence.
b. Increase the sample size.
c. Decrease the level of confidence.
d. Decrease the sample size.
37. A 95% confidence interval indicates that:
a. 95% of the possible sample means (same-size samples) will be included by the interval.
b. 95% of the intervals constructed using this process based on same-sized samples from this population will include the sample mean.
c. 95% of the possible population means will be included by the interval.
d. 95% of the intervals constructed using this process based on same-sized samples from this population will include the population mean.
38. A nationally distributed college newspaper conducts a survey among students nationwide every year. This year, responses from a simple random sample of 204 college students to the question "About how many CDs do you own?" resulted in a sample mean of 72.8. Based on data from previous years, the editors of the newspaper will assume a population standard deviation of 7.2. What is a 95% confidence interval for the population mean number of CDs owned by all college students? Show your work.
a. (65.6, 80.0) b. (71.8, 73.8) c. (72.0, 73.6) d. (72.3, 73.3)
39. The whole point of doing a confidence interval is:
a. To get information about the sample statistic.
b. To do a census.
c. To estimate the standard deviation.
d. To estimate the population parameter