Stat 1350 - Elementary Statistics

Graded Homework Assignment 9Unit 8, Lessons 1-2

Lesson 1

1. Choose one of the five experiments given below to complete:

Nickels spinning. Hold a nickel upright on its edge under your forefinger on a hard surface, then snap it with your other forefinger so that it spins for some time before falling. Based on 40 spins, estimate the probability of heads.

Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about 1/2 because the coin has two faces. Such opinions are not always correct. The previous exercise asked you to spin a nickel rather than toss itthat changes the probability of a head. Now try another variation. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. What is the probability that it falls with heads upward? Make at least 40 trials to estimate the probability of a head.

Random digit's. The table of random digits (TableA) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the first 400 digits in the table are 0s? This proportion is an estimate, based on 400 repetitions, of the true probability, which in this case is known to be 0.1.

How many tosses to get a head? When we toss a penny, experience shows that the probability (long-term proportion) of a head is close to 1/2. Suppose now that we toss the penny repeatedly until we get a head. What is the probability that the first head comes up in an odd number of tosses (1, 3, 5, and so on)? To find out, repeat this experiment 40 times, and keep a record of the number of tosses needed to get a head on each of your 40 trials.

(a) From your experiment, estimate the probability of a head on the first toss. What value should we expect this probability to have?

(b) Use your results to estimate the probability that the first head appears on an odd-numbered toss.

Tossing a thumbtack. Toss a thumbtack on a hard surface 50 times. How many times did it land with the point up? What is the approximate probability of landing point up?

Complete all of the following problems:

2. From words to probabilities. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.)

00.010.30.60.991

(a) This event is impossible. It can never occur._________________

(b) This event is certain. It will occur on every trial.______________

(c) This event is very unlikely, but it will occur once in a while in a long sequence of trials.__________

(d) This event will occur more often than not.________________

3. Winning a baseball game. Over the period from 1965 to 2011 the champions of baseball's two major leagues won 63% of their home games during the regular season. At the end of each season, the two league champions meet in the baseball World Series. Would you use the results from the regular season to assign probability 0.63 to the event that the home team wins a World Series game? Explain your answer.

4. Marital status. The probability that a randomly chosen 65-year-old woman is divorced is about 0.14. This probability is a long-run proportion based on all the millions of women aged 65. Let's suppose that the proportion stays at 0.14 for the next 45 years. Bridget is now 20 years old and is not married.

(a) Bridget thinks her own chances of being divorced at age 65 are about 5%. Explain why this is a personal probability.

(b) Give some good reasons why Bridget's personal probability might differ from the proportion of all women aged 65 who are divorced.

(c) You are a government official charged with looking into the impact of the Social Security system on retirement-aged divorced women. You care only about the probability 0.14, not about anyone's personal probability. Why?

5. Personal probability? When there are few data, we often fall back on personal probability. There had been just 24 space shuttle launches, all successful, before the Challenger disaster in January 1986. The shuttle program management thought the chances of such a failure were only 1 in 100,000.

(a) Suppose 1 in 100,000 is a correct estimate of the chance of such a failure. If a shuttle was launched every day, about how many failures would one expect in 300 years?

(b) Give some reasons why such an estimate is likely to be too optimistic.

6. Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. The winning number is essentially a four-digit group from a table of random digits. You win if your choice matches the winning digits, in exact order. The winnings are divided among all players who matched the winning digits. That suggests a way to get an edge.

(a) The winning number might be, for example, either 2873 or 9999. Explain why these two outcomes have exactly the same probability. (It is 1 in 10,000.)

(b) If you asked many people which outcome is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this chapter to say which one and to explain why.

7. An eerie coincidence? An October 6, 2002, ABC News article reported that the winning New York State lottery numbers on the one-year anniversary of the attacks on America were 911. Should this fact surprise you? Explain your answer.

8. In the long run. Probability works not by compensating for imbalances but by overwhelming them. Suppose that the first 10 tosses of a coin give 10 tails and that tosses after that are exactly half heads and half tails. (Exact balance is unlikely, but the example illustrates how the first 10 outcomes are swamped by later outcomes.) What is the proportion of heads after the first 10 tosses? What is the proportion of heads after 100 tosses if half of the last 90 produce heads (45 heads)? What is the proportion of heads after 1000 tosses if half of the last 990 produce heads? What is the proportion of heads after 10,000 tosses if half of the last 9990 produce heads?

9. Snow coming. A meteorologist, predicting below-average snowfall this winter, says, "First, in looking at the past few winters, there has been above-average snowfall. Even though we are not supposed to use the law of averages, we are due." Do you think that "due by the law of averages" makes sense in talking about the weather? Explain.

10. Reacting to risks. The probability of dying if you play high school football is about 10 per million each year you play. The risk of getting cancer from asbestos if you attend a school in which asbestos is present for 10 years is about 5 per million. If we ban asbestos from schools, should we also ban high school football? Briefly explain your position.

11. What probability doesn't say. The probability of a head in tossing a coin is 1/2. This means that as we make more tosses, the proportion of heads will eventually get close to 0.5. It does not mean that the count of heads will get close to 1/2 the number of tosses. To see why, imagine that the proportion of heads is 0.49 in 100 tosses, 0.493 in 1000 tosses, 0.4969 in 10,000 tosses, and 0.49926 in 100,000 tosses of a coin. How many heads came up in each set of tosses? How close is the number of heads to half the number of tosses?

Lesson 2

12. Causes of death. Government data assign a single cause for each death that occurs in the United States. The data show that the probability is 0.34 that a randomly chosen death was due to heart disease, and 0.23 that it was due to cancer. What is the probability that a death was due either to heart disease or to cancer? What is the probability that the death was due to some other cause?

13. Our next president? A Gallup Poll on Presidents Day 2008 interviewed a random sample of 1007 adult Americans. Those in the sample were asked which former president they would like to bring back as the next president if they could. Here are the results:

These proportions are probabilities for the random phenomenon of choosing an adult American at random and asking her or his opinion.

(a) What must be the probability that the person chosen selects someone other than John F. Kennedy, Ronald Reagan, or Abraham Lincoln? Why?

(b) The event "I would select either John F. Kennedy or Ronald Reagan" contains the first two outcomes. What is its probability?

14. Rolling a die. Figure18.5 displays several assignments of probabilities to the six faces of a die. We can learn which assignment is actually correct for a particular die only by rolling the die many times. However, some of the assignments are not legitimate assignments of probability. That is, they do not obey the rules. Which are legitimate and which are not? In the case of the illegitimate models, explain what is wrong.

Figure18.5Four probability models for rolling a die, for Exercise18.7.

15. High school academic rank. Select a first-year college student at random and ask what his or her academic rank was in high school. Here are the probabilities, based on proportions from a large sample survey of first-year students:

(a) What is the sum of these probabilities? Why do you expect the sum to have this value?

(b) What is the probability that a randomly chosen first-year college student was not in the top 20% of his or her high school class?

(c) What is the probability that a first-year student was in the top 40% in high school?

16. Birth order. A couple plan to have three children. List the possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a boy. Allarrangements are (approximately) equally likely.Select the correct answer for each problem below.

(a) Write down all the arrangements of the sexes of three children. What is the probability of any one of these arrangements?

(b) What is the probability that the couple's children are 2 girls and 1 boy?

17. Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.

(a) What is the probability of any one of the 38 possible outcomes? Explain your answer.

(b) If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning?

(c) The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains all multiples of 3, that is, 3, 6, 9,..., 36. You place a "column bet" that wins if any of these numbers comes up. What is your probability of winning?

18. Colors of M&MS. If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made.

(a) Here are the probabilities of each color for a randomly chosen plain M&M:

What must be the probability of drawing a blue candy?_______________________________

(b) The probabilities for peanut M&MS are a bit different. Here they are:

What is the probability that a peanut M&M chosen at random is blue?______________________

(c) What is the probability that a plain M&M is any of red, yellow, or orange? What is the probability that a peanut M&M has one of these colors?

19. Legitimate probabilities? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer.

(a) When a coin is spun, P(H) = 0.55 and P(T) = 0.45.___________________

(b) When two coins are tossed, P(HH) = 0.4, P(HT) = 0.4, P(TH) = 0.4, and P(TT) = 0.4.__________

(c) Plain M&M'S have not always had the mixture of colors given in Exercise 18(a). In the past there were no red candies and no blue candies. Tan had probability 0.10 and the other four colors had the same probabilities that are given in Exercise 18(a)._____________________________________

20. Airplane safety. Suppose that 68% of all adults think that airplanes would be safer places if airline passengers were banned from carrying on board any luggage, including purses, computers, and briefcases. An opinion poll plans to ask an SRS of 1023 adults about airplane safety. The proportion of the sample who think that airplanes would be safer if passengers were banned from carrying on board any luggage, including purses, computers, and briefcases, will vary if we take many samples from this same population. The sampling distribution of the sample proportion is approximately Normal with mean 0.68 and standard deviation about 0.015. Sketch this Normal curve and use it to answer the following questions.

(a) What is the probability that the poll gets a sample in which more than 71% of the people think that airplanes would be safer if passengers were banned from carrying on board any luggage, including purses, computers, and briefcases?

(b) What is the probability of getting a sample that misses the truth (68%) by 3.0% or more?

21.In the setting of Exercise20, what is the probability of getting a sample in which more than 70% think that airplanes would be safer if passengers were banned from carrying on board any luggage, including purses, computers, and briefcases? (Use TableB.)

22. Applying to college. You ask an SRS of 1500 college students whether they applied for admission to any other college. Suppose that in fact 35% of all college students applied to colleges besides the one they are attending. (That's close to the truth.) The sampling distribution of the proportion of your sample who say "Yes" is approximately Normal with mean 0.35 and standard deviation 0.01. Sketch this Normal curve and use it to answer the following questions.

(a) Explain in simple language what the sampling distribution tells us about the results of our sample.

(b) What percentage of many samples would have a larger than 0.37? (Use the 68-95-99.7 rule.) Explain in simple language why this percentage is the probability of an outcome larger than 0.37.

(c) What is the probability that your sample will have a less than 0.33?

(d) Use Rule D: what is the probability that your sample result will be either less than 0.33 or greater than 0.35?