Suppose that potholes along the motorway north of Las Angles have been found to occur according to a Poisson process with rate parameter λ.

(a) Out of 46 three kilometer stretches of motorway, the average number of potholes was found to be 6.2. Find the maximum likelihood estimate for the rate parameter, λˆ MLE, and the standard deviation of the estimate, SD(λˆ MLE).

(b) Using the maximum likelihood estimate as the rate parameter, what is the probability that a randomly chosen three kilometer stretch of motorway has more than five potholes?

(c) If three randomly chosen, nonoverlapping, stretches of the freeway are checked by inspectors, what is the probability that only one of them has more than five potholes?

In parts (b) and (c), please identify the relevant variable and its distribution, and express the required probability in terms of this variable before calculating your answer. You may use the formula, tables or a calculator to find the actual probability. For example (in a different scenario) you might write: “Let N be the number of dry wells before drilling a well that finds oil, then N ∼ NB(n = 1, p = 0.25). We want Pr(N < 5) = ....”

A CBS News/New York Times survey found that 97% of Americans believe that texting while driving should be outlawed (CBS News website, January 5, 2015). a. For a sample of 10 Americans, what is the probability that at least 8 say that they believe texting while driving should be outlawed? Use the binomial distribution probability function to answer this question (to 4 decimals). b. For a sample of 100 Americans, what is the probability that at least 95 say that they believe texting while driving should be outlawed? Use the normal approximation of the binomial distribution to answer this question (to 4 decimals). Use “Continuity correction factor” method. Use Table 1 in Appendix B. c. As the number of trials in a binomial distribution application becomes large, what is the advantage of using the normal approximation of the binomial distribution to compute probabilities? d. When the number of trials for a binominal distribution application becomes large, would developers of statistical software packages prefer to use the binomial distribution probability function shown in Section 5.5 or the normal approximation of the binomial distribution discussed in Section 6.3? Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Answer ASAP: Probability and Statistics question

On average, 2.4 car accidents occur on certain highway each month (which is assumed to always have 30 days). Answer the following questions:

(a) What is the probability of having at least 1 accident on the highway next month?

Hint: Let LaTeX: XX denote the number of car accidents on the highway next month. What distribution does LaTeX: X X have?

(b) What is the probability that there will be no accident during the first 15 days of next month?

Hint: Let LaTeX: YYdenote the waiting time (in months) for the first accident that will occur on the highway next month. What does the question imply about LaTeX: YY?

(c) Given that there is no accident on the highway during the first 15 days of next month, what is the probability that there is no accident during the next 10 days of the month?

(d) Bonus question (5 extra points). Considering next year, let LaTeX: ZZ denote the number of months in which there will be no car accidents on the highway. What distribution does LaTeX: ZZ have? Be sure to specify both the distribution name and parameter value(s).

A CBS News/New York Times survey found that 97% of Americans believe that texting while driving should be outlawed (CBS News website, January 5, 2015).

a. For a sample of 10 Americans, what is the probability that at least 8 say that they believe texting while driving should be outlawed? Use the binomial distribution probability function to answer this question (to 4 decimals).

b. For a sample of 100 Americans, what is the probability that at least 95 say that they believe texting while driving should be outlawed? Use the normal approximation of the binomial distribution to answer this question (to 4 decimals). Use “Continuity correction factor” method. Use Table 1 in Appendix B.

c. As the number of trials in a binomial distribution application becomes large, what is the advantage of using the normal approximation of the binomial distribution to compute probabilities?

d. When the number of trials for a binominal distribution application becomes large, would developers of statistical software packages prefer to use the binomial distribution probability function shown in Section 5.5 or the normal approximation of the binomial distribution discussed in Section 6.3?

Explain.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

After years of rapid growth, illegal immigration into the United States has declined, perhaps owing to the recession and increased border enforcement by the United States (Los Angeles Times, September 1, 2010). While its share has declined, California still accounts for 23% of the nation’s estimated 11.1 million undocumented immigrants. [You may find it useful to reference the z table.]

a. In a sample of 50 illegal immigrants, what is the probability that more than 20% live in California? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

b. In a sample of 200 illegal immigrants, what is the probability that more than 20% live in California? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. Comment on the reason for the difference between the computed probabilities in parts a and b.

• As the sample number increases, the probability of more than 20% also increases, due to the lower z value and decreased standard error.

• As the sample number increases, the probability of more than 20% also increases, due to the lower z value and increased standard error.

A) According to the National Institute of Health, 34% of adults in the United States are overweight. Suppose we sample 20 adults. Let X denote the number of overweight people among them.

What is the probability that more than 6 of the adults will be overweight? Round to four decimal places.

B) According to the National Institute of Health, 34% of adults in the United States are overweight. Suppose we sample 20 adults. Let X denote the number of overweight people among them.

What is the probability that exactly 5 of the adults will be overweight? Round to four decimal places.

C) The weight of infants at a New York hospital has a mean of 7.5 pounds and a standard deviation of 0.95 pounds. Weights are approximately normally distributed.

What is the probability that a randomly selected infant will have a weight between 6.8 pounds and 7.9 pounds? Round to four decimal places.

D) The lengths of pregnancies are normally distributed with a mean of 248 days and a standard deviation of 15 days. If a pregnant woman is randomly selected, what is the probability that her pregnancy lasts more than 275 days? Round to four decimal places.

Shandelle rolls a pair of fair dice and sums the number of spots that appear on the up faces. She then flips a fair coin the number times associated with the sum of the spots. For example, if she rolled a 3 and a 4, then she flips the fair coin 7 times. If the coin flipping part of the random experiment yielded an equal number of heads and tails, find the probability that she rolled an 8 on the dice rolling part of the random experiment.

1/The heights of adult men in America are normally distributed, with a mean of 69.2 inches and a standard deviation of 2.65 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.5 inches and a standard deviation of 2.51 inches.
a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)?
z =
b) What percentage of men are SHORTER than 6 feet 3 inches? Round to nearest tenth of a percent.
%
c) If a woman is 5 feet 11 inches tall, what is her z-score (to two decimal places)?
z =
d) What percentage of women are TALLER than 5 feet 11 inches? Round to nearest tenth of a percent.
e) Who is relatively taller: a 6'3" American man or a 5'11" American woman? Defend your choice in a meaningful sentence.

Suppose that about 84% of graduating students attend their graduation. A group of 35 students is randomly chosen, and let X be the number of students who attended their graduation.

Please show the following answers to 4 decimal places.

1. What is the distribution of X? X ~ ? B U N  (,)
2. What is the probability that exactly 25 number of students who attended their graduation in this study?
3. What is the probability that less than 25 number of students who attended their graduation in this study?
4. What is the probability that at least 25 number of students who attended their graduation in this study?
5. What is the probability that between 22 and 26 (including 22 and 26) number of students who attended their graduation in this study?
6. 3/According to the American Red Cross, 11% of all Connecticut residents have Type B blood. A random sample of 20 Connecticut residents is taken.

   X=X=the number of CT residents that have Type B blood, of the 20 sampled.
   What is the expected value of the random variable XX?
   2.28 2.26 2.04 2.2 1.9 2.08

7. 4/The owner of a small deli is trying to decide whether to discontinue selling magazines. He suspects that only 10.7% of his customers buy a magazine and he thinks that he might be able to use the display space to sell something more profitable. Before making a final decision, he decides that for one day he will keep track of the number of customers that buy a magazine. Assuming his suspicion that 10.7% of his customers buy a magazine is correct, what is the probability that exactly 6 out of the first 12 customers buy a magazine?

Assume a binomial probability distribution has p = 0.80 and n = 400. (a) What are the mean and standard deviation? (Round your answers to two decimal places.) mean standard deviation (b) Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain. No, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because np < 5 and n(1 − p) < 5. No, because np < 5 and n(1 − p) < 5. Yes, because n ≥ 30. (c) What is the probability of 300 to 310 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.) (d) What is the probability of 330 or more successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.) (e) What is the advantage of using the normal probability distribution to approximate the binomial probabilities? The advantage would be that using the normal probability distribution to approximate the binomial probabilities increases the number of calculations. The advantage would be that using the the normal probability distribution to approximate the binomial probabilities reduces the number of calculations. The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations more accurate. The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations less accurate. How would you calculate the probability in part (d) using the binomial distribution. (Use f(x) to denote the binomial probability function.) P(x ≥ 330) = f(0) + f(1) + + f(329) + f(330) P(x ≥ 330) = f(331) + f(332) + f(333) + f(334) + + f(400) P(x ≥ 330) = 1 − f(329) − f(330) − f(331) − f(332) − − f(400) P(x ≥ 330) = f(330) + f(331) + f(332) + f(333) + + f(400) P(x ≥ 330) = f(0) + f(1) + + f(328) + f(329)

Assume a binomial probability distribution has

p = 0.60

and

n = 300.

(a)

What are the mean and standard deviation? (Round your answers to two decimal places.)

mean standard deviation

(b)

Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain.

No, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because n ≥ 30.     Yes, because np ≥ 5 and n(1 − p) ≥ 5. No, because np < 5 and n(1 − p) < 5. Yes, because np < 5 and n(1 − p) < 5.

(c)

What is the probability of 160 to 170 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(d)

What is the probability of 190 or more successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(e)

What is the advantage of using the normal probability distribution to approximate the binomial probabilities?

The advantage would be that using the normal probability distribution to approximate the binomial probabilities increases the number of calculations. The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations less accurate.     The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations more accurate. The advantage would be that using the the normal probability distribution to approximate the binomial probabilities reduces the number of calculations.

How would you calculate the probability in part (d) using the binomial distribution. (Use f(x) to denote the binomial probability function.)

P(x ≥ 190) = f(191) + f(192) + f(193) + f(194) +    + f(300)

P(x ≥ 190) = f(190) + f(191) + f(192) + f(193) +    + f(300)

    

P(x ≥ 190) = f(0) + f(1) +    + f(188) + f(189)

P(x ≥ 190) = f(0) + f(1) +    + f(189) + f(190)

P(x ≥ 190) = 1 − f(189) − f(190) − f(191) − f(192) −    − f(300)