stats chapter 5

discrete

(a) number of traffic fatalities per year in the state of Florida

continuous

(a) speed of an airplane

continuous

(b) age of a college professor chosen at random

continuous

(b) distance a golf ball travels after being hit with a driver

discrete

(c) number of books in the college bookstore

continuous

(c) time required to drive from home to college on any given day

discrete

(d) number of ships in Pearl Harbor on any given day

continuous

(d) weight of a football player chosen at random

discrete

(e) number of lightning strikes in Rocky Mountain National Park on a given day

continuous

(e) your weight before breakfast each morning

.125

A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.) (a) Find the probability of getting exactly three heads.

.375

A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.) (b) Find the probability of getting exactly two heads.

.5

A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.) (c) Find the probability of getting two or more heads.

.125

A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in the binomial probability distribution table. (Enter your answers to three decimal places.) (d) Find the probability of getting exactly three tails.

Yes. 120 is more than 2.5 standard deviations above the expected value.

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.

Yes. 40 is more than 2.5 standard deviations below the expected value.

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. (b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain.

No. 70 to 90 observations is within 2.5 standard deviations of the expected value.

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. (c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

This distribution is symmetric.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution. (a) The probability of success is p = 0.50.

This distribution is skewed to the right.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution. (b) The probability of success is p = 0.25.

This distribution is skewed to the left.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution. (c) The probability of success is p = 0.75.

The distributions are mirror images of each other.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution. (d) What is the relationship between the distributions shown in parts (b) and (c)?

Skewed to the left, since p > 0.50.

Consider a binomial distribution with n = 5 trials. Use the probabilities given in the Binomial Probability Distribution table in the Appendix to make histograms showing the probabilities of r = 0, 1, 2, 3, 4, and 5 successes for each of the following. Comment on the skewness of each distribution. (e) If the probability of success is p = 0.73, do you expect the distribution to be skewed to the right or to the left? Why?

The expected value is higher for the first distribution.

Consider two binomial distributions, with n trials each. The first distribution has a higher probability of success on each trial than the second. How does the expected value of the first distribution compare to that of the second?

two; success or failure

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

geometric distribution

For a binomial experiment, what probability distribution is used to find the probability that the first success will occur on a specified trial?

No. A binomial experiment requires that the probability of success be the same for each trial.

In a binomial experiment, is it possible for the probability of success to change from one trial to the next? Explain.

Yes. The five trials are independent, have only two outcomes, and have the same P(success); n = 5, r = 2, p = 1/6

In a carnival game, there are six identical boxes, one of which contains a prize. A contestant wins the prize by selecting the box containing it. Before each game, the old prize is removed and another prize is placed at random in one of the six boxes. Is it appropriate to use the binomial probability distribution to find the probability that a contestant who plays the game five times wins exactly twice? Check each of the requirements of a binomial experiment and give the values of n, r,and p.

The trials are independent The probability of success on each trial is constant. A fixed number of trials repeated under identical conditions. The trials have exactly two outcomes.

List the criteria for a binomial experiment. (Select all that apply.)

No, n = 50 is not large enough.

Suppose we have a binomial experiment with 50 trials, and the probability of success on a single trial is 0.02. Is it appropriate to use the Poisson distribution to approximate the probability of two successes? Explain.

Yes, np < 10 and n ≥ 100.

Suppose we have a binomial experiment, and the probability of success on a single trial is 0.02. If there are 150 trials, is it appropriate to use the Poisson distribution to approximate the probability of three successes? Explain.

A description of all the values of the random variable x. Associated probabilities for each value of x. Each probability takes on values between 0 and 1 inclusive. The summation of the probabilities equal 1.

What are the requirements for a probability distribution? (Select all that apply.)

The outcome of one trial does not affect the probability of success on any other trial.

What does it mean to say that the trials of an experiment are independent?

the average number of successes

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

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

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

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

What does the random variable of a binomial experiment measure?

expected value; 𝜆

When using the Poisson distribution, which parameter of the distribution is used in probability computations? What is the symbol used for this parameter?