Chapter 5, 6 and 7: Review Questions: STAT/MATH Consider the experiment of rolling a fair die twice. Find the indicated probabilities.

Transcription

1 Chapter5 Chapter 5, 6 and 7: Review Questions: STAT/MATH Consider the experiment of rolling a fair die twice. Find the indicated probabilities. (a) One of the dice is a 4. (b) Sum of the dice equals 4. (c) Neither dice is a Suppose that A and B are any two events, with P(A) = 0.2, P(B) = 0.6 and P(B A) = (a) Find P(A B). (b) Are A and B independent? Explain (c) Are A and B mutually exclusive? Explain 3. Suppose that A and B are any two events, with P(A) = 0.3, P(B) = 0.4 and P(B A) = 0.1 (a) Find P(A B). (b) Are A and B independent? Explain (c) Are A and B mutually exclusive? Explain 4. Suppose that A and B are any two events, with P(A) = 0.5, P(B) = 0.4 and P(A B) = (d) Find P(A B). (e) Find P(B A) (f) Find P( [A U B] C ) (g) Are A and B independent? Explain (h) Are A and B mutually exclusive? Explain

2 5. Consider the experiment of rolling a fair die twice. Find the probabilities. (a) Pair of 1 s (b) One of the dice shows 5 6. Consider the experiment of rolling a fair die twice. Find the probabilities. (c) Sum is less than 7 (d) Same number on both dice 7. Suppose P(A) = 0.54, P(B) = 0.45, and P(AB) = (a) Find the conditional probability that B occurs given that A occurs. (b) Are A and B independent? Why or Why not? (c) Are A and B mutually exclusive? Why or Why not? 8. Sample space for an experiment is given by S = { e1, e2, e3, e4}. Are the following assignments of probabilities permissible, mark yes or no? (i) P ( e 1 ) = 0. 5, P ( e 2 ) = 0. 1, P ( e 3 ) = 0, P ( e 4 ) = 0. 4 (ii) P ( e 1 ) = 0. 6, P ( e 2 ) = 0. 5, P ( e 3 ) = 0. 5, P ( e 4 ) = 0. 4 (iii) P ( e 1 ) = 0. 3, P ( e 2 ) = 0. 1, P ( e 3 ) = 0. 1 P ( e 4 ) = Two fair die are tossed. (a) List the elements for the events A = [ sum = 4] B = [Same number on each die] C = [sum is greater than 9] (b) Obtain P(A), P(B) and P(C) 10. Construct a tree-diagram for the experiment of tossing a coin three times. 11. What is an equally likely experiment?

3 12. Consider the experiment of drawing a single card from a deck of 52 cards. Find the probability of observing the following events. a. Heart b. Face card (king, queen, or jack) c. Seven d. Red card e. Seven of hearts f. Red queen 13. Consider the experiment of drawing a card at random from a shuffled deck of 52 cards. (a) What is the probability of drawing a red card or a king? (b) What is the probability of drawing a king and a heart? (c) What is the probability of drawing a heart and a spade? 14. A committee of four people has to be formed from a class of 10 students. Find the number of ways this committee can be formed. 15. A traveling salesman plans to visit Atlanta, Miami, Alabama, Tampa, and Birmingham in one week. How many different routes are possible? 16. How many random samples of size 2 can be chosen from a population of size 10? 17. Find the value of the combination 10 C Find the value of the permutation 5 P How many distinct strings of letters can be made using all the letters in the word PROBABILITY? Chapter 6 1. It is reported that 70% of high school females engaged in aerobics or dancing for vigorous physical activity. A random sample of 8 students is taken this year. (a) Find the probability that the sample contains 3 females who engaged in aerobics or dancing. (b) Repeat part(a) using binomial formula (not the table) (c) Find the probability that the sample contains at most 2 females who engaged in aerobics or dancing. (d) Find the probability that the sample contains at least 5 females who engaged in aerobics or dancing. (e) What is the most likely number of females in the sample who engaged in aerobic or dancing? (f) If you find 2 out of the 8 students engaged in aerobic or dancing, would that be considered as unusual?

4 2. It is reported that 55% of high school females engaged in aerobics or dancing for vigorous physical activity. A random sample of 12 students is taken this year. (a) Find the probability that the sample contains 6 females who engaged in aerobics or dancing. (b) Find the probability that the sample contains at most 3 females who engaged in aerobics or dancing. (c) Find the probability that the sample contains at least 11 females who engaged in aerobics or dancing. (d) What is the most likely number of females in the sample who engaged in aerobic or dancing? (e) If you find 11 out of the 12 students engaged in aerobic or dancing, would that be considered as unusual? 3. Participants from the National Health and Nutrition Examination Survey, , had a mean BMI(Body Mass Index) of 28.3 with a standard deviation of 6.0. Assume BMI is normally distributed. (a) What is the probability that a randomly selected participant had a BMI measure between 25 and 32? (b) What is the probability that a randomly selected participant had a BMI measure below 28.3? (c) Bottom 25% would be eligible for a food allowance from the government. What is the cutoff BMI value to be eligible for the allowance? 4. Find the indicated probability for the standard normal distribution (a) P ( Z 2.87) (b) P ( 1.64 < Z < 1.99) (a) Find the Z value with an area of under the standard normal curve to its right. (b) Find the Z value with an area of under the standard normal curve to its left. (c) Find the Z value with an area of under the standard normal curve to its left. 6. The distribution of heights (X) of American women aged 18 to 24 is normally distributed with mean 65.5 inches and standard deviation 2.5 inches. (a) What is the probability that a randomly selected woman is between 60 and 64 inches tall?

5 (b)tallest 25% will be edible to be selected to the basketball-pool. What is the cutoff to be selected to the basketball-pool? 7. (a) Find the Z-value with an area of under the standard normal curve to its right. (b) Evaluate P(Z > -1.54). Find the 27 th percentile of the Z distribution. 8. Find the indicated probability for the standard normal (a) P(Z > 1.67) (b) P(-1.75< Z < 1.96) P(Z < -2.95) (d) Find the 80 th percentile of the Z distribution. Find the Z-value with an area of under the standard normal curve to its right. 9.Find the mean and the standard deviation of the following distribution. x P(x) Chipper Jones was the Major League Baseball league leader in 2008 according to batting average. For the 2008 season, the percentage of Jones hits that resulted in 1, 2, 3, or 4 (home run) bases was as follows: Calculate the standard deviation of X. X = number of bases P(X) Find the Z-value with an area of under the standard normal curve to its left. 12. Find the Z-value with an area of under the standard normal curve to its right. 13. Assume that the random variable X is normally distributed with mean μ = 120 and standard deviation σ = 13. Find P(100 < X < 155). 14. #41, #42 on page 325 from the book.

6 Chapter7 1. Determine whether the sampling distribution of x is normal, approximately normal, or unknown. (a) Heights of American women aged 18 to 24 are distributed with mean 65.5 inches and standard deviation 2.5 inches and the distribution is unknown. A random sample of 40 is selected. (b) Average salaries for full-time associate professors in United States doctoral departments of psychology are normally distributed with a mean of $71955 and a standard deviation of $ A sample of size 20 is taken. (c) A study involving the economic burden of congestive heart failure found that the lengths of hospital stays for patients are not normally distributed with a mean of 7.8 days and a standard deviation of 9.1 days. A random sample of size 27 is taken. (d) According to the Census Bureau s 2002 American Community Survey, the average travel time to work of workers 16 years and over living in Boston, MA, who did not work at home, was 28.2 minutes with a standard deviation of 0.79 minutes. A sample of 35 commute times is taken. (e) Suppose that in a particular area, the average length of a hospital stay is 4.8 days with a standard deviation of 3.0 days. In a sample of 50 patients, compute the probability the mean stay is above 5 days. 2. It is reported that in the state of New York, approximately 35% lived within one mile of a hazardous waste site. (a) Would p based on a random sample of 200 residents have approximately a normal distribution? (b) What is the smallest value for n for which the sampling distribution of p is approximately normal? What are the mean value and standard deviation of p based on a sample of size 400? (a) When n = 400, what is P [ 0.25 < pˆ < 0.45]?

7 3. If possible find the indicated probability. If not possible, explain why not (a) The gestation time for human babies is assumed to be normally distributed with a mean of 278 days and a standard deviation of 12 days. Suppose we take a sample of size 15. What is the probability that the sample mean will be more than 270 days? (b) Assume the fill amount of bottles of a soft drink is distributed with a mean of 2.0 liters and standard deviation of 0.04 liter. If we take a random sample of 12 bottles, what is the probability we will obtain a sample mean of liters or more? (c) A study involving the economic burden of congestive heart failure found that the lengths of hospital stays for patients had a mean of 7.8 days with a standard deviation of 9.1 days. A random sample of size 49 is taken. Find P(7.5< x < 8.5). 4. A random sample of size 100 is taken from a population having a mean 25 and a standard deviation of 4. The shape of the population is unknown. (a) Find the mean and the standard deviation of X. (b) Find P [ X > 25.65] 5. The gestation time for human babies is assumed to be normally distributed with a mean of 278 days and a standard deviation of 12 days. Suppose we take a sample of size 15. What is the probability that the sample mean will be more than 270 days? In 2005, 25% of all female homicide victims were murdered by an intimate (Source: U.S. Department of Justice). (a) Find the minimum sample size n * that produces a sampling distribution of p ˆ that is approximately normal. (b) Find the probability that in a sample of 25 female homicide victims, fewer than 7 were murdered by an intimate. 5. It is reported that in the state of New York, approximately 35% lived within one mile of a hazardous waste site. (a) What is the smallest value for n for which the sampling distribution of p is approximately normal? (b) What are the mean value and standard deviation of p based on a sample of size 50? When n = 50, what is P [ 0.30 < pˆ < 0.55]? 6. What sample size is necessary to use the normal approximation to the binomial when p = 0.90?

8 7. Assume that body temperatures of healthy adults are normally distributed with a mean of 98.2 F and a standard deviation of 0.62 F. Suppose we take a sample of eight healthy adults. What is the probability that their mean body temperature will be greater than 98.4?