Stats 201 Ch 6 Probability Questions

. Under what circumstances is the normal distribution an accurate approximation of the binomial distribution?

1. When pn and qn are both greater than 10

1. For a normal distribution with a mean of |a = 60 and a standard deviation of ct = 12, find each probability value requested. a. p(X > 66) b. p(X < 75) c. p(X < 57) d. p (48 < X < 72)

1. a. p = 0.3085 b. p = 0.8944 c. p = 0.4013 d. p = 0.6826

1. A local hardware store has a "Savings Wheel" at the checkout. Customers get to spin the wheel and, when the wheel stops, a pointer indicates how much they will save. The wheel can stop in any one of 50 sections. Of the sections, 10 produce 0% off, 20 sections are for 10% off, 10 sections for 20%, 5 for 30%, 3 for 40%, 1 for 50%, and 1 for 100% off. Assuming that all 50 sections are equally likely, a. What is the probability that a customer's purchase will be free (100% off)? b. What is the probability that a customer will get no savings from the wheel (0% off)? c. What is the probability that a customer will get at least 20% off?

1. a. p = 1/50 = 0.02 b. p = 10/50 = 0.20 c. p = 20/50 = 0.40

1. Find the proportion of a normal distribution that corresponds to each of the following sections: a. z < 0.25 b. z > 0.80 c. z < -1.50 d. z > -0.75

1. a. p ~ 0.5987 b. p = 0.2119 c. p = 0.0668 d. p = 0.7734

1. A survey of the students in a psychology class revealed that there were 19 females and 8 males. Ot the 19 females, only 4 had no brothers or sisters, and 3 of the males were also the only child in the household. If a student is randomly selected from this class, a. What is the probability of obtaining a male? b. What is the probability of selecting a student who has at least one brother or sister? c. What is the probability of selecting a female who has no siblings?

1. b. 20/27 c. 4/27

10. Find the z-score location of a vertical line that separates a normal distribution as described in each of the following. a. 20% in the tail on the left b. 40% in the tail on the right c. 75% in the body on the left d . 99% in the body on the right

10. a. z = -0.84 c. z = 0.67 b. z = 0.25 d. z = -2.33

11. Find the z-score boundaries that separate a normal distribution as described in each of the following. a. The middle 20% from the 80% in the tails. b. The middle 50% from the 50% in the tails. c. The middle 95% from the 5% in the tails. d. The middle 99% from the 1 % in the tails

11. a. z = ±0.25 c. z = ±1.96 b. z = ±0.67 d. z = ±2.58

12. For a normal distribution with a mean of p = 80 and a standard deviation of ct = 20, find the proportion of the population corresponding to each of the following scores. a. Scores greater than 85. b. Scores less than 100. c. Scores between 70 and 90.

12. a. p(z > 0.25) = 0.4013 b. p(z < 1.00) = 0.8413 c. p(-0.50 < z < 0.50) = 0.3830

13. A normal distribution has a mean of p = 50 and a standard deviation of ct = 12. For each of the following scores, indicate whether the tail is to the right or left of the score and find the proportion of the distribution located in the tail. a. X = 53 b. X = 44 c. X = 68 d. X = 38

13. a. tail to the right, p = 0.4013 b. tail to the left, p = 0.3085 c. tail to the right, p = 0.0668 d. tail to the left, p = 0.1587

14. IQ test scores are standardized to produce a normal distribution with a mean of p = 100 and a standard deviation of cr = 15. Find the proportion of the population in each of the following IQ categories. a. Genius or near genius: IQ greater than 140 b. Very superior intelligence: IQ between 120 and 140 c. Average or normal intelligence: IQ between 90 and 109

14. a. z = 2.67, p = 0.0038 b. p(1.33 < z < 2.67) = 0.0880 c. p(-0.67 < z < 0.60) = 0.4743

15. The distribution of scores on the SAT is approximately normal with a mean of p = 500 and a standard deviation of cr = 100. For the population of students who have taken the SAT, a. What proportion have SAT scores greater than 700? b. What proportion have SAT scores greater than 550? c. What is the minimum SAT score needed to be in the highest 10% of the population? d. If the state college only accepts students from the top 60% of the SAT distribution, what is the minimum SAT score needed to be accepted?

15. a. z = 2.00, p = 0.0228 b. z = 0.50, p = 0.3085 c. z = 1.28, X = 628 d. z = -0.25, X = 475

16. The distribution of SAT scores is normal with |i = 500 and ct = 100. a. What SAT score, X value, separates the top 15% of the distribution from the rest? b. W hat SAT score, X value, separates the top 10% of the distribution from the rest? c. What SAT score, X value, separates the top 2% of the distribution from the rest?

16. a. z = 1.04, X = 604 b. z = 1.28, X = 628 c. z = 2.05, X = 705

17. A recent newspaper article reported the results of a survey of well-educated suburban parents. The responses to one question indicated that by age 2, children were watching an average of |i = 60 minutes of television each day. Assuming that the distribution of television-watching times is normal with a standard deviation of ct = 20 minutes, find each of the following proportions. a. What proportion of 2-year-old children watch more than 90 minutes of television each day? b. What proportion of 2-year-old children watch less than 20 minutes a day?

17. a. p(z > 1.50) = 0.0668 b. p(z < -2.00) = 0.0228

18. Information from the Department of Motor Vehicles indicates that the average age of licensed drivers is p = 45.7 years with a standard deviation of cr = 12.5 years. Assuming that the distribution of drivers' ages is approximately normal, a. What proportion of licensed drivers are older than 50 years old? b. What proportion of licensed drivers are younger than 30 years old?

18. a. z = 0.34, p = 0.3669 b. z = -1.26, p = 0.1038

19. A consumer survey indicates that the average household spends p = $185 on groceries each week. The distribution of spending amounts is approximately normal with a standard deviation of cr = $25. Based on this distribution, a. What proportion of the population spends more than $200 per week on groceries? b. What is the probability of randomly selecting a family that spends less than $150 per week on groceries? c. How much money do you need to spend on groceries each week to be in the top 20% of the distribution?

19. a. z = 0.60, p = 0.2743 b. z = -1.40, p = 0.0808 c. z = 0.84, X = $206 or more

2. In the game Rock-Paper-Scissors, the probability that both players will select the same response and tie is p = y, and the probability that they will pick different responses is p = §. If two people play 72 rounds of the game and choose their responses randomly, what is the probability that they will choose the same response (tie) more than 28 times?

2. With• p = j and1 q = y, the binomial distribution is normal with p = 24 and a = 4;2 p(X > 28.5) = p(z >1.13) = 0.1292.

2. For a normal distribution, find the z-score location that divides the distribution as follows: a. Separate the top 20% from the rest. b. Separate the top 60% from the rest. c. Separate the middle 70% from the rest

2. a. c = 0.84 b. z = -0 .2 5 c. z = — 1.04 and + 1.04

2. A psychology class consists of 14 males and 36 females. If the professor selects names from the class list using random sampling, a. What is the probability that the first student selected will be a female? b. If a random sample of n = 3 students is selected and the first two are both females, what is the probability that the third student selected will be a male?

2. a. p = 36/50 = 0.72 b. p = 14/50 = 0.28 (Remember, a random sample requires replacement.)

2. Aj ar contains 10 red marbles and 30 blue marbles. a. If you randomly select 1 marble from the jar, what is the probability of obtaining a red marble? b. If you take a random sample of n = 3 marbles from the jar and the first two marbles are both blue, what is the probability that the third marble will be red?

2. a. p = f = 0.25 b- P = 45 = 0.25. Remember that random sampling requires sampling with replacement.

2. Scores on the Mathematics section of the SAT Reasoning Test form a normal distribution with a mean of |i = 500 and a standard deviation of ct = 100. a. If the state college only accepts students who score in the top 60% on this test, what is the minimum score needed for admission? b. What is the minimum score necessary to be in the top 10% of the distribution? c. What scores form the boundaries for the middle 50% of the distribution?

2. a. z = -0.25; X = 475 b. z = 1.28; X = 628 c. z = ±0.67; X =433 and X =567

20. Over the past 10 years, the local school district has measured physical fitness for all high school freshmen. During that time, the average score on a treadmill endurance task has been p = 19.8 minutes with a standard deviation of cr = 7.2 minutes. Assuming that the distribution is approximately normal, find each of the following probabilities. a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? In symbols. p(X > '25) = ? b. What is the probability of randomly selecting a student with a time greater than 30 minutes? In symbols, p(X > 30) = ? c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what proportion of the freshmen would fail?

20. a. p(z > 0.72) = 0.2358 b. p(z > 1.42) = 0.0778 c. p(z < -1.36) = 0.0869

21. Rochester, New York, averages p = 21.9 inches of snow for the month of December. The distribution of snowfall amounts is approximately normal with a standard deviation of ct = 6.5 inches. This year, a local jewelry store is advertising a refund of 50% off of all purchases made in December, if Rochester finishes the month with more than 3 feet (36 inches) 198 CHAPTER 6 PROBABILITY of total snowfall. What is the probability that the jewelry store will have to pay off on its promise?

21. p(X > 36) = p(z > 2.17) = 0.0150 or 1.50%

22. A multiple-choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers, a. What is the probability of guessing correctly for any question? b. On average, how many questions would a student get correct for the entire test? c. What is the probability that a student would get more than 15 answers correct simply by guessing? d. What is the probability that a student would get 15 or more answers correct simply by guessing?

22. a. p = ¼ b. μ = 12 c. σ = √9 = 3 and for X = 15.5, z = 1.17, and p = 0.1210 d. For X = 14.5, z = 0.83, and p = 0.2033

23. A true/false test has 40 questions. If a students is simply guessing at the answers, a. What is the probability of guessing correctly for any one question? b. On average, how many questions would the student get correct for the entire test? c. What is the probability that the student would get more than 25 answers correct simply by guessing? d. What is the probability that the student would get 25 or more answers correct simply by guessing?

23. a. p = ½ b. μ = 20 c. σ = √10 = 3.16 and for X = 25.5, z = 1.74, and p = 0.0409 d. For X = 24.5, z = 1.42, and p = 0.0778

24. A roulette wheel has alternating red and black numbered slots into one of which the ball finally stops to determine the winner. If a gambler always bets on black to win, what is the probability of winning at least 24 times in a series of 36 spins? (Note that at least 24 wins means 24 or more.)

24. p = q = 1/2, and with n = 36 the normal approximation has μ = 18 and σ = 3. Using the lower real limit of 23.5, p(X > 23.5) = p(z > 1.83) = 0.0336.

25. One test for ESP involves using Zener cards. Each card shows one of five different symbols (square, circle, star, cross, wavy lines), and the person being tested has to predict the shape on each card before it is selected. Find each of the probabilities requested for a person who has no ESP and is just guessing. a. What is the probability of correctly predicting 20 cards in a series of 100 trials? b. What is the probability of correctly predicting more than 30 cards in a series of 100 trials? c. What is the probability of correctly predicting 50 or more cards in a series of 200 trials?

25. a. With five options, p = 1/5 for each trial. μ = 20 and σ = 4 For X = 20, z = ±0.13 and p = 0.1034. b. For X = 30.5, z = 2.63 and p = 0.0043. c. μ = 40 and σ = 5.66 For X = 49.5, z = 1.68 and p = 0.0465

26. A trick coin has been weighted so that heads occurs with a probability ofp = j , and /j(tails) = j . If you toss this coin 72 times, a. How many heads would you expect to get on average? b. What is the probability of getting more than 50 heads? c. What is the probability of getting exactly 50 heads?

26. a. = pn = 48 b. z = 0.63, p = 0.2643 c. p(0.38 < z < 0.63) = 0.0877

27. For a balanced coin: a. What is the probability of getting more than 30 heads in 50 tosses? b. What is the probability of getting more than 60 heads in 100 tosses? c. Parts a and b both asked for the probability of getting more than 60% heads in a series of coin tosses ( ^ = = 60%). Why do you think the two probabilities are different?

27. a. With n = 50 and p = q = 1/2, you may use the normal approximation with μ = 25 and σ = 3.54. Using the upper real limit of 30.5, p(X > 30.5) = p(z > 1.55) = 0.0606. b. The normal approximation has μ = 50 and σ = 5. Using the upper real limit of 60.5, p(X > 60.5) = p(z > 2.10) = 0.0179. c. Getting 60% heads with a balanced coin is an unusual event for a large sample. Although you might get 60% heads with a small sample, you should get very close to a 50-50 distribution as the sample gets larger. With a larger sample, it becomes very unlikely to get 60% heads.

28. A national health organization predicts that 20% of American adults will get the flu this season. If a sample of 100 adults is selected from the population, a. What is the probability that at least 25 of the people will be diagnosed with the flu? (Be careful: "at least 25" means "25 or more.") b. What is the probability that fewer than 15 of the people will be diagnosed with the flu? (Be careful: "fewer than 15" means " 14 or less."

28. a. μ = 20 and σ = 4 For X = 24.5, z = 1.13, and p = 0.1292. b. For X = 14.5, z = -1.38, and p = 0.0838.

3. W hat are the two requirem ents that must be satisfied for a random sample?

3. The two requirements for a random sample are: (1) each individual has an equal chance of being selected, and (2) if more than one individual is selected, the probabilities must stay constant for all selections.

4. What is sampling with replacement, and why is it used?

3. The two requirements for a random sample are: (1) each individual has an equal chance of being selected, and (2) if more than one individual is selected, the probabilities must stay constant for all selections.

3. If you toss a balanced coin 36 times, you would expect, on the average, to get 18 heads and 18 tails. What is the probability of obtaining exactly 18 heads in 36 tosses?

3. X = 18 is an interval with real limits of 17.5 and 18.5. The real limits correspond to z = ±0.17, and a probability ofp = 0.1350.

3. What is the probability of selecting a score greater than 45 from a positively skewed distribution with p = 40 and ct = 10? (Be careful.)

3. You cannot obtain the answer. The unit normal table cannot be used to answer this question because the distribution is not normal.

3. Suppose that you are going to select a random sample of /; = 1 score from the distribution in Figure 6.2. Find the following probabilities: a. p(X > 2) b. p(X > 5) c. p(X < 3)

3. a. p=7/10 b- P = 0.1 c- P = m = 0.3

5. Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the tail is on the right or left side of the line and find the proportion in the tail. a. z = 2.00 b. z = 0.60 c. z = -1.30 d. z = -0.30

5. a. tail to the right, p = 0.0228 b. tail to the right, p = 0.2743 c. tail to the left, p = 0.0968 d. tail to the left, p = 0.3821

6. Draw a vertical line through a normal distribution for each of the following z-score locations. Determine whether the body is on the right or left side of the line and find the proportion in the body. a. z = 2.20 b. z = 1.60 c. z = -1.50 d. z = -0.70

6. a. body to the left, p = 0.9861 b. body to the left, p = 0.9452 c. body to the right, p = 0.9332 d. body to the right, p = 0.7580

7. Find each of the following probabilities for a normal distribution. a. p(z > 0.25) b. p(z > -0.75) c. p(z < 1.20) d. p(z < — 1.20)

7. a. p(z > 0.25) = 0.4013 c. p(z < 1.20) = 0.8849 b. p(z > -0.75) = 0.7734 d. p(z < -1.20) = 0.1151

8. What proportion of a normal distribution is located between each of the following z-score boundaries? a. z = -0.50 and z = +0.50 b. z = -0.90 and z = +0.90 c. z = - 1.50 and z

8. a. p = 0.3830 b. p = 0.6318 c. p = 0.8664

9. Find each of the following probabilities for a normal distribution. a. pi-0.25 < z < 0.25) b. p(-2.00 < z < 2.00) c. p (-0.30 < z < 1.00) d . p ( - 1.25 < z < 0.25)

9. a. p = 0.1974 c. p = 0.4592 b. p = 0.9544 d. p = 0.4931

3. The tail will be on the right-hand side of a normal distribution for any positive z-score. (True or false)

True