Assistance for AMU MATH 120 Intro to statistics Homework week 4 American Military University is available at Domyclass

1. Which of the following are continuous variables, and which are discrete?

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

- continuous
- discrete

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

- continuous
- discrete

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

- continuous
- discrete

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

- continuous
- discrete

(e) your weight before breakfast each morning

- continuous
- discrete

2. Consider the probability distribution shown below.

X | 0 | 1 | 2 |

P(x) | 0.05 | 0.30 | 0.65 |

Compute the expected value of the distribution. (Enter a number.)

Compute the standard deviation of the distribution. (Enter a number. Round your answer to four decimal places.)

3. What is the income distribution of super shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars.

Income range | 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55 or more |

Midpoint X | 10 | 20 | 30 | 40 | 50 | 60 |

Percent of super shoppers | 20% | 14% | 20% | 16% | 20% | 10% |

(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain.

- Yes. The events are indistinct and the probabilities sum to less than 1.
- No. The events are indistinct and the probabilities sum to more than 1.
- Yes. The events are distinct and the probabilities do not sum to 1.
- No. The events are indistinct and the probabilities sum to 1.
- Yes. The events are distinct and the probabilities sum to 1.

(b) Use a histogram to graph the probability distribution of part (a). (Select the correct graph.)

(c) Compute the expected income μ of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)

μ = thousands of dollars

(d) Compute the standard deviation σ for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)

σ = thousands of dollars

4. 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 6hour period. The percentage data are the percentages of fishermen who caught x fish in a 6hour period while fishing from shore.

x | 0 | 1 | 2 | 3 | 4 or more |

% | 43% | 35% | 15% | 6% | 1% |

- Convert the percentages to probabilities and make a histogram of the probability distribution. (Select the correct graph.)

(b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6hour period. (Enter a number. 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 6hour period. (Enter a number. Round your answer to two decimal places.)

(d) Compute μ, the expected value of the number of fish caught per fisherman in a 6hour period (round 4 or more to 4). (Enter a number. Round your answer to two decimal places.) μ = fish

(e) Compute σ, the standard deviation of the number of fish caught per fisherman in a 6hour period (round 4 or more to 4). (Enter a number. Round your answer to three decimal places.) σ = fish

5. Jim is a 60yearold Anglo male in reasonably good health. He wants to take out a $50,000 term (i.e., straight death benefit) life insurance policy until he is 65. The policy will expire on his 65th birthday. The probability of death in a given year is provided.

X=age | 60 | 61 | 62 | 63 | 64 |

P(death at this age) | 0.01033 | 0.01321 | 0.01681 | 0.01912 | 0.02236 |

Jim is applying to Big Rock Insurance Company for his term insurance policy.

(a) What is the probability that Jim will die in his 60th year? (Enter a number. Enter your answer to five decimal places.)

Using this probability and the $50,000 death benefit, what is the expected cost to Big Rock Insurance (in dollars)? (Enter a number. Round your answer to two decimal places.)

(b) What is the expected cost to Big Rock Insurance for years 61, 62, 63, and 64 (in dollars)? (For each answer, enter a number. Round your answers to two decimal places.)

year 61 $

year 62 $

year 63 $

year 64 $

What would be the total expected cost to Big Rock Insurance over the years 60 through 64 (in dollars)? (Enter a number. Round your answer to two decimal places.)

(c) If Big Rock Insurance wants to make a profit of $700 above the expected total cost paid out for Jim’s death, how much should it charge for the policy (in dollars)? (Enter a number. Round your answer to two decimal places.)

(d) If Big Rock Insurance Company charges $5000 for the policy, how much profit does the company expect to make (in dollars)? (Enter a number. Round your answer to two decimal places.)

6. Consider a binomial experiment with n = 9 trials where the probability of success on a single trial is p = 0.10. (For each answer, enter a number. Round your answers to three decimal places.)

(a) Find P(r = 0).

- Find P(r ≥ 1) by using the complement rule.

7. Sociologists say that 90% of married women claim that their husband’s mother is the biggest bone of contention in their marriages (sex and money are lowerrated areas of contention). Suppose that ten married women are having coffee together one morning. Find the following probabilities. (For each answer, enter a number. Round your answers to three decimal places.)

(a) All of them dislike their motherinlaw.

(b) None of them dislike their motherinlaw.

(c) At least eight of them dislike their motherinlaw.

(d) No more than seven of them dislike their motherinlaw.

8. Suppose approximately 75% of all marketing personnel are extroverts, whereas about 70% of all computer programmers are introverts. (For each answer, enter a number. Round your answers to three decimal places.)

(a) At a meeting of 15 marketing personnel, what is the probability that 10 or more are extroverts?

What is the probability that 5 or more are extroverts?

What is the probability that all are extroverts?

(b) In a group of 5 computer programmers, what is the probability that none are introverts?

What is the probability that 3 or more are introverts?

What is the probability that all are introverts?

9. 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.

- Would it be unusual to have more than 120 successes out of 200 trials? Explain.
- Yes. 120 is more than 2.5 standard deviations above the expected value.
- Yes. 120 is more than 2.5 standard deviations below the expected value.
- No. 120 is less than 2.5 standard deviations above the expected value.
- No. 120 is less than 2.5 standard deviations below the expected value.

- Would it be unusual to have fewer than 40 successes out of 200 trials? Explain.
- Yes. 40 is more than 2.5 standard deviations above the expected value.
- Yes. 40 is more than 2.5 standard deviations below the expected value.
- No. 40 is less than 2.5 standard deviations above the expected value.
- No. 40 is less than 2.5 standard deviations below the expected value.

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

- No. 90 observations is more than 2.5 standard deviations above the expected value.
- Yes. 70 to 90 observations is within 2.5 standard deviations of the expected value.
- No. 70 to 90 observations is within 2.5 standard deviations of the expected value.
- Yes. 70 observations is more than 2.5 standard deviations below the expected value.

10.The qualitycontrol inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of eight syringes taken from the batch. Suppose the batch contains 5% defective syringes.

(a) Make a histogram showing the probabilities of r = 0, 1, 2, 3, …, 7 and 8 defective syringes in a random sample of eight syringes. (Select the correct graph.)

(b) Find μ (in terms of the number of syringes). (Enter a number. Enter your answer to two decimal places.)

μ = syringes

What is the expected number of defective syringes the inspector will find? (Enter a number. Enter your answer to two decimal places.)

syringes

(c) What is the probability that the batch will be accepted? (Enter a number. Round your answer to three decimal places.)

(d) Find σ (in terms of the number of syringes). (Enter a number. Round your answer to three decimal places.)

σ = syringes

11 . Innocent until proven guilty? In Japanese criminal trials, about 95% of the defendants are found guilty. In the United States, about 60% of the defendants are found guilty in criminal trials. (Source: The Book of Risks, by Larry Laudan, John Wiley and Sons) Suppose you are a news reporter following eight criminal trials. (For each answer, enter a number.)

(a) If the trials were in Japan, what is the probability that all the defendants would be found guilty? (Round your answer to three decimal places.)

What is this probability if the trials were in the United States? (Round your answer to three decimal places.)

(b) Of the eight trials, what is the expected number of guilty verdicts in Japan? (Round your answer to two decimal places.)

What is the expected number in the United Sates? (Round your answer to two decimal places.)

What is the standard deviation in Japan? (Round your answer to two decimal places.) verdicts

What is the standard deviation in the United States? (Round your answer to two decimal places.)

12. Use the geometric probability distribution to solve the following problem.

On the leeward side of the island of Oahu, in a small village, about 89% of the residents are of Hawaiian ancestry. Let n = 1, 2, 3, … represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village.

(a) Write out a formula for the probability distribution of the random variable n. (Enter a mathematical expression.)

P(n) =

(b) Compute the probabilities that n = 1, n = 2, and n = 3. (For each answer, enter a number. Round your answers to three decimal places.)

P(1) =

P(2) =

P(3) =

(c) Compute the probability that n ≥ 4. Hint: P(n ≥ 4) = 1 − P(n = 1) − P(n = 2) − P(n = 3). (Enter a number. Round your answer to three decimal places.)

(d) What is the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry? Hint: Use μ for the geometric distribution and round. (Enter a number. Round your answer to the nearest whole number.)