Probability Distribution and Binomial Distribution, Exams of Communication

Download Probability Distribution and Binomial Distribution and more Exams Communication in PDF only on Docsity!

x x

x

COMM215 ASSIGNEMENT CHAPTER 6-7 Concordia University

1. Sound City sells the ClearTone-400 satellite car radio. For this radio, historical sales records over the last 100 weeks show 2 weeks with no radios sold, 18 weeks with one radio sold, 18 weeks with two radios sold, 46 weeks with three radios sold, 9 weeks with four radios sold, and 7 weeks with five radios sold. Calculate μ x , σ 2 , and σ , of x, the number of ClearTone-400 radios sold at Sound City during a week using the estimated probability distribution. (Round your answers to 2 decimal places.)

Explanation (a) μ x = 0 × .02 + 1 × .18 + 2 × .18 + 3 × .46 + 4 × .09 + 5 × .07 = 2. (b) σ 2 = (0-2.63)^2 × .02 + (1-2.63)^2 × .18 + (2-2.63)^2 × .18 + (3-2.63)^2 × .18 + (4-2.63)^2 ×. 46 + (5-2.63)^2 ×. = .1383 + .4782 + .0714 + .0630 + .1689 + .3932 = 1.

(c) σ x = SQRT(1.31) = 1.

2. Of all individual tax returns, 37 percent include errors made by the taxpayer. If IRS examiners are assigned randomly selected returns in batches of 12, find the mean and standard deviation for the number of erroneous returns per batch.

Explanation Mean = np = 12(.37) = 4. Standard Deviation = √( npq ) = √*(12)(.37)(.63)+ = 1.

3. The probability distribution of X is

X P(X) 3 1/ 4 1/ 5 3/ 6 3/

What is the expected value of X?

Explanation

E [ X ] = 3 ( 1 / 8 )+ 4 ( 1 / 8 )+ 5 ( 3 / 8 )+ 6 ( 3 / 8 )= 40 / 8 = 5.

4. If p = .1 and n = 5, then the corresponding binomial distribution is .

Explanation Most of the values for this binomial distribution are on the left-hand side of the graph.

5. Using the following probability distribution table of the random variable x , what is the probability of x = 3?

X P ( X ) 0 5/ 1 4/ 2 1/ 3

Explanation

All values of P ( X ) need to sum to 1, so 5/15 + 4/15 + 1/15 = 10/15 means P ( X = 3) = 5/15.

6. Which of the following is a valid probability value for a discrete random variable?

left skewed

P ( X ) = n !/[ x !( n − x )!] × px (1 − p ) n − x , for x = 4, when p = .20 and n = 30.

9. X has the following probability distribution P ( X ).

X 1 2 3 4

P(X) 0.1 0.5 0.2 0.

Compute the variance value of X.

Explanation E[X] = (1)(.1) + (2)(.5) + (3)(.2) + (4)(.2) = 2. σ 2 xσ2x = (1 − 2.5)^2 (.1) + (2 − 2.5)^2 (.5) + (3 − 2.5)^2 (.2) + (4 − 2.5)^2 (.2) =.

10. The number of ways to arrange x successes among n trials is equal to

Explanation

N is the number of trials in an experiment and x is the number of successes in that experiment.

11. A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur fewer than 3 times?

Explanation Σ P ( X ) = n !/[ x !( n − x )!] × px (1 − p ) n−x , for x = 1, 3, 5; or look up in binomial table where x = 1 or 3 or 5 when p = .5 and n = 10.

12. If x is a binomial random variable, then the standard deviation of x is given by

Explanation

Where n is the number of trials, p is the number of successes and q is the number of failures.

13. According to a survey of adults, 64 percent have money in a bank savings account. If we were to survey 50 randomly selected adults, find the mean number of adults who would have bank savings accounts.

Explanation Mean = np = 50(.64) = 32

14. The United States Golf Association requires that the weight of a golf ball must not exceed 1.62 oz. The association periodically checks golf balls sold in the United States by sampling specific brands stocked by pro shops. Suppose that a manufacturer claims that no more than 4 percent of its brand of golf balls exceed 1. oz. in weight. Suppose that 24 of this manufacturer's golf balls are randomly selected, and let x denote the number of the 24 randomly selected golf balls that exceed 1.62 oz. Refer to the Binomial table given below.

Excel Output of the Binomial Distribution with n = 24, p = .04, and q =.

Binomial distribution with n = 24 and p =.

x P ( X = x ) 0 0. 1 0. 2 0. 3 0. 4 0.

P ( x ≥ 2) = 1 – P ( x ≤ 1) = 1 – .7509 =.

15. The mean of the binomial distribution is equal to Multiple Choice

Explanation P ( X = 5) = (.4)^5 =.

17. Suppose that x is a binomial random variable with n = 5, p = .45, and q = .55.

(b) For each value of x , calculate p ( x ). (Round final answers to 4 decimal places.)

(c) Find P ( x = 3). (Round final answer to 4 decimal places.)

(d) Find P ( x ≤ 3). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(e) Find P ( x < 3). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(f) Find P ( x ≥ 4). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(g) Find P ( x > 2). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

p.

18. Which one of the following statements is not an assumption of the binomial distribution?

Explanation All trials are independent and you do not replace once you have a success or failure.

19. In the most recent election, 19 percent of all eligible college students voted. If a random sample of 20 students were surveyed, find the probability that none of the students voted.

Explanation P ( X ) = n !/[ x !( n − x )!] × px (1 − p ) n−x , for x = 0 when p = .19 and n = 20.

20. A car wash loses $30 on rainy days and makes $ on days when it does not rain. If the probability of rain is .15, calculate expected profit for the car wash.

Explanation μx = (−30)(.15) + (120)(.85) = −4.50 + 102 = 97.

21. Which of the following statements about the binomial distribution is not correct?

Explanation

The random variable of interest needs to be discrete.

22. The internal auditor for your company believes that 10 percent of your invoices contain errors. To check this theory, 20 invoices are randomly selected, and 5 are found to have errors. What is the probability that of the 20 invoices selected, 5 or more would contain errors if the theory is valid?

Explanation P ( x ≥ 5) = .0319 + .0089 + .0020 + .0004 + .0001 =.

23. In the most recent election, 19 percent of all eligible college students voted. If a random sample of 20 students were surveyed, find the probability that exactly half voted in the election. Multiple Choice -. -. Correct -. -. Explanation

P ( X ) = n !/[ x !( n − x )!] × px (1 − p ) n−x , for x = 10, when p = .19 and n = 20.

24. A fair die is rolled 36 times. What is the standard deviation of the even number (2, 4, or 6) outcomes? Multiple Choice

• 18
• 9
• 3 Correct

Explanation

Binomial standard deviation = σ = √( npq ) = √*(36)(.5)(.5)+ = √9 = 3

25. Determine the probability that a 3 will appear twice, if a single fair die is rolled 10 times.

(2) Use the binomial formula to calculate the probability that at least three customers make a purchase. (Round your answer to 4 decimal places.)

(3) Use the binomial formula to calculate the probability that two or fewer customers make a purchase. (Round your answer to 4 decimal places.)

(4) Use the binomial formula to calculate the probability that at least one customer makes a purchase. (Round your answer to 4 decimal places.)

Explanation

(1) P ( x = 5) =.

(2) P ( x ≥ 3) = 1 – P ( x ≤ 2) = 1 – .4618 =.

(3) P ( x ≤ 2) =.

(4) P ( x ≥ 1) = 1 – P ( x = 0) = 1 – .0308 =.

29. One die is thrown. What is the expected value of the number of dots on the top face of the die?

Explanation

E [ X ] = 1 ( 1 / 6 )+ 2 ( 1 / 6 )+ 3 ( 1 / 6 )+ 4 ( 1 / 6 )+ 5 ( 1 / 6 )+ 6 ( 1 / 6 )

30. The binomial distribution is characterized by situations that are analogous to

31. For a binomial process, the probability of success is 40 percent and the number of trials is 5. Find the variance.

Explanation σ^2 = (5)(.4)(.6) = 1.

32. A study conducted by a local university found that 25 percent of college freshmen support increased spending on environmental issues. If 6 college freshmen are randomly selected, find the probability that fewer than 4 support increased spending on environmental issues.

Explanation Σ P ( X ) = n !/[ x !( n − x )!] × px (1 − p ) n−x , for x = 0, 1, 2, 3 when p = .25 and n = 6.

33. According to data from the state blood program, 40 percent of all individuals have group A blood. Suppose that of six randomly selected individuals, three have group A blood. Would you believe the data from the state blood program?

Explanation Yes the probability is greater than 0.05 and therefore we do not reject the null hypothesis.

34. Which of the following is not a discrete random variable?

Explanation It could take someone infinite number of minutes to finish the 1 mile.

35. If p = .5 and n = 4, then the corresponding binomial distribution is.

Explanation It is similar to a bell curve.

36. A vaccine is 95 percent effective. What is the probability that it is not effective for more than 1 out of 20 individuals?

Multiple Choice

-. Correct -. -. -. Explanation

P ( X ≥ 2) = 1 − [ P ( X = 0) + p ( X = 1)] = 1 − [.3585 + .3774] =.

37. If you were asked to play a game in which you tossed a fair coin three times and were given $2 for every head you threw, how much would you expect to win on average?

Explanation

Expected return = ( 0 )( 1 / 8 )+($2)( 1 )( 3 / 8 )+($2)( 2 )( 3 / 8 )+($2)( 3 )( 1 / 8 )= 3.

38. A total of 50 raffle tickets are sold for a contest to win a car. If you purchase one ticket, what are your odds

left skewed

Explanation

Probability of losing = 1 − probability of winning = 1 − 1/50 = 49/50.

39. When p = .5, the binomial distribution will be symmetric.

Explanation This looks similar to a normal distribution.

40. A lawyer believes that the probability is .3 that she can win a discrimination suit. If she wins the case, she will make $400,000; but if she loses, she gets nothing. Assume that she has to spend $75,000 preparing the case. What is her expected gain?

Explanation μx = .7(−75,000) + .3(325,000) = −52,500 + 97,500 = 45,

ASSIGNEMENT CHAPTER 7

1. An apple juice producer buys all his apples from a conglomerate of apple growers in one northwestern state. The amount of juice obtained from each of these apples is approximately normally distributed with a mean of 2.25 ounces and a standard deviation of .15 ounce. Between what two values (in ounces), symmetrically distributed around the population mean, will 80 percent of the apples fall?