Probability and Statistics: Week 4 Exercise

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(−2.14 ≤ z ≤ −0.33) =

.3545

Find z such that 6% of the standard normal curve lies to the left of z. (Round your answer to three decimal places.)

1.555

What does the expected value of a binomial distribution with n trials tell you?

the average number of successes

For a binomial experiment, how many outcomes are possible for each trial? What are the possible outcomes?

two; success or failure

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z ≥ 1.33) =

.0918

Consider the probability distribution shown below. x 0 1 2 P(x) 0.05 0.20 0.75 Compute the expected value of the distribution. Compute the standard deviation of the distribution. (Round your answer to four decimal places.)

1.7 .5568

Look at the two normal curves in the figures below. Which has the larger standard deviation? What is the mean of the curve in Figure (a)? What is the mean of the curve in Figure (b)?

Figure (a) 10 4

(e) Consider two normal curves. If the first one has a larger mean than the second one, must it have a larger standard deviation as well? Explain your answer.

No. The values of μ and σ are independent.

A particular lake is known to be one of the best places to catch a certain type of fish. In this table, x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore. x 0 1 2 3 4 or more % 42% 34% 14% 9% 1%

Answers below

What does the random variable for a binomial experiment of n trials measure?

The random variable measures the number of successes out of n trials.

Raul received a score of 80 on a history test for which the class mean was 70 with a standard deviation of 10. He received a score of 80 on a biology test for which the class mean was 70 with standard deviation 3. On which test did he do better relative to the rest of the class?

biology test

The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About 20% of all adults deliberately do a one-time fling and feel no guilt about it! In a group of eight adult friends, what is the probability of the following? (Round your answers to three decimal places.) (a) no one has done a one-time fling (b) at least one person has done a one-time fling (c) no more than two people have done a one-time fling

(a).168 (b).832 (c).798

Let r be a binomial random variable representing the number of successes out of n trials. (a) Explain why the sample space for r consists of the set {0, 1, 2, ..., n} and why the sum of the probabilities of all the entries in the entire sample space must be 1. (b) Explain why P(r ≥ 1) = 1 − P(0). (c) Explain why P(r ≥ 2) = 1 − P(0) − P(1). (d) Explain why P(r ≥ m) = 1 − P(0) − P(1) − ... − P(m − 1) for 1 ≤ m ≤ n.

(a) (b) (c) (d)

Richard has just been given a 6-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all six questions, find the indicated probabilities. (Round your answers to three decimal places.) (a) What is the probability that he will answer all questions correctly? (b) What is the probability that he will answer all questions incorrectly? (c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table. Then use the fact that P(r ≥ 1) = 1 − P(r = 0). Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference? (d) What is the probability that Richard will answer at least half the questions correctly?

(a) (b) (c) (d) (e)

A normal distribution has μ = 30 and σ = 5. (a) Find the z score corresponding to x = 25. (b) Find the z score corresponding to x = 42. (c) Find the raw score corresponding to z = −3. (d) Find the raw score corresponding to z = 1.5.

(a) -1 (b) 2.4 (c) 15 (d) 37.5

Consider a binomial experiment with n = 6 trials where the probability of success on a single trial is p = 0.05. (Round your answers to three decimal places.) (a) Find P(r = 0). (b) Find P(r ≥ 1) by using the complement rule.

(a) .735 (b) .265

Assuming that the heights of college women are normally distributed with mean 65 inches and standard deviation 3.4 inches, answer the following questions. (Hint: Use the figure below with mean μ and standard deviation σ.) (a) What percentage of women are taller than 65 inches? (b) What percentage of women are shorter than 65 inches? (c) What percentage of women are between 61.6 inches and 68.4 inches? (d) What percentage of women are between 58.2 and 71.8 inches?

(a) 50 (b) 50 (c) 68 (d) 95

(a) What percentage of the area under the normal curve lies to the left of μ? (b) What percentage of the area under the normal curve lies between μ − σ and μ + σ? (c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?

(a) 50 (b) 68 (c) 99.7

What percentage of the area under the normal curve lies as given below? (a) to the right of μ (b) between μ - 2σ and μ + 2σ (c) to the right of μ + 3σ (Use 2 decimal places.)

(a) 50 (b) 95 (c) .15

The incubation time for a breed of chicks is normally distributed with a mean of 23 days and standard deviation of approximately 3 days. Look at the figure below and answer the following questions. If 1000 eggs are being incubated, how many chicks do we expect will hatch in the following time periods? (Note: In this problem, let us agree to think of a single day or a succession of days as a continuous interval of time. Assume all eggs eventually hatch.) (a) in 17 to 29 days (b) in 20 to 26 days (c) in 23 days or fewer (d) in 14 to 32 days

(a) 950 (b) 680 (c) 500 (d) 997

Are your finances, buying habits, medical records, and phone calls really private? A real concern for many adults is that computers and the Internet are reducing privacy. A survey conducted by Peter D. Hart Research Associates for the Shell Poll was reported in USA Today. According to the survey, 37% of adults are concerned that employers are monitoring phone calls. Use the binomial distribution formula to calculate the probability of the following. (a) Out of four adults, none is concerned that employers are monitoring phone calls. (Round your answer to three decimal places.) (b) Out of four adults, all are concerned that employers are monitoring phone calls. (Round your answer to three decimal places.) (c) Out of four adults, exactly two are concerned that employers are monitoring phone calls. (Round your answer to three decimal places.)

(a).158 (b).019 (c).326

Trevor is interested in purchasing the local hardware/sporting goods store in the small town of Dove Creek, Montana. After examining accounting records for the past several years, he found that the store has been grossing over $850 per day about 50% of the business days it is open. Estimate the probability that the store will gross over $850 for the following. (Round your answers to three decimal places.) (a) at least 3 out of 5 business days (b) at least 6 out of 10 business days (c) fewer than 5 out of 10 business days (d) fewer than 6 out of the next 20 business days If the outcome described in part (d) actually occurred, might it shake your confidence in the statement p = 0.50? Might it make you suspect that p is less than 0.50? Explain. (e) more than 17 out of the next 20 business days If the outcome described in part (e) actually occurred, might you suspect that p is greater than 0.50? Explain.

(a).499 (b).377 (c) (d).02 (e).000

Consider a binomial distribution of 200 trials with expected value 80 and standard deviation of about 6.9. Use the criterion that it is unusual to have data values more than 2.5 standard deviations above the mean or 2.5 standard deviations below the mean to answer the following questions. (a) Would it be unusual to have more than 120 successes out of 200 trials? Explain. (b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain. (c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

(a)Yes. 120 is more than 2.5 standard deviations above the expected value. (b)Yes. 40 is more than 2.5 standard deviations below the expected value. (c)No. 70 to 90 observations is within 2.5 standard deviations of the expected value.

Consider each distribution. Determine if it is a valid probability distribution or not, and explain your answer. (a)x 0 1 2 P(x) 0.22 0.64 0.14 (b)x 0 1 2 P(x) 0.22 0.64 0.15

(a)Yes. The probabilities sum to 1. (b)No. The probabilities do not sum to 1.

Which of the following are continuous variables, and which are discrete? (a) number of traffic fatalities per year in the state of Florida (b) distance a golf ball travels after being hit with a driver (c) time required to drive from home to college on any given day (d) number of ships in Pearl Harbor on any given day (e) your weight before breakfast each morning

(a)discrete (b)continuous (c)continuous (d)discrete (e)continuous

(b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period. (Round your answer to two decimal places.) (c) Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period. (Round your answer to two decimal places.) (d) Compute μ, the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Round your answer to two decimal places.) (e) Compute σ, the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Round your answer to three decimal places.)

(b) .58 (c) .24 (d) .93 (e) 1.003

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 39; σ = 15 P(50 ≤ x ≤ 70) =

.2130

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z ≤ −0.12) =

.4522

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(−1.12 ≤ z ≤ 2.73) =

.8654

Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(z ≤ 1.28) =

.8997