3 A construct a probabilty histogram of the following distri

3. A. construct a probabilty histogram of the following distributuion x-0 p(x)- 0.1 / x-2 p(x)-0.3 / x-4 p(x)-0.4/ x-5 p(x)-0.2

B. determine the expected value of the discrete random variable

C. Using the formula for the mean and standard deviation, determine the standard deviation of the distribution.

4. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of n, p,q, and list the possible values of the random variable x. If x is not a binomial random variable, explain why.

4A. Cyanosis is a condition of having bluish skin due to insufficent oxygen in the blood. About 80% of babies born with cynanosis recover fully. A hospital is caring for five babies born with cyanosis. The random variable represents the number of babies tht fully recover.

4B. A politcal polling organization calls 1012 people and asks, \"Do you approve, disapprove, or have no opinion of the way the president is handling his job?\" The random variable is the number of people who approve of the way the president is handling his job.

6. In Pittsburgh, Pennsylvania about 56% of the days in a year are cloudy. Find the mean,variance, and standard deviation for the number od cloudy days in the month of june.

9. A study found that the mean migration distance of the green turtle was 2200 kilometers and the standard deviation was 625 kilometers. Assuming that the distances are normally distributed, find the probability that a randomly selected green turtle migrates a distnace of 9A. more than 1900 kilometers 9B. between 2000 kilometers and 2500 kilometers

11. Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about$34 and the estimated standard deviation is about $9.

(a) Consider a random sample of n = 60 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

The sampling distribution of x is not normal.The sampling distribution of x is approximately normal with mean _x = 34 and standard error _x = $1.16.    The sampling distribution of x is approximately normal with mean _x = 34 and standard error _x = $9.The sampling distribution of x is approximately normal with mean _x = 34 and standard error _x = $0.15.

Is it necessary to make any assumption about the x distribution? Explain your answer.

It is not necessary to make any assumption about the x distribution because n is large.It is necessary to assume that x has an approximately normal distribution.    It is necessary to assume that x has a large distribution.It is not necessary to make any assumption about the x distribution because is large.

(b) What is the probability that x is between $32 and $36? (Round your answer to four decimal places.)


(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $32 and $36? (Round your answer to four decimal places.)


(d) In part (b), we used x, the average amount spent, computed for 60 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

The mean is larger for the x distribution than it is for the x distribution.The sample size is smaller for the x distribution than it is for the x distribution.    The x distribution is approximately normal while the x distribution is not normal.The standard deviation is larger for the x distribution than it is for the x distribution.The standard deviation is smaller for the x distribution than it is for the x distribution.

In this example, x is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

The central limit theorem tells us that the standard deviation of the sample mean is much smaller than the population standard deviation. Thus, the average customer is more predictable than the individual customer.The central limit theorem tells us that small sample sizes have small standard deviations on average. Thus, the average customer is more predictable than the individual customer.   

13. Is p an unbiased estimator for p when np > 5 and nq > 5? Recall that a statistic is an unbiased estimator of the corresponding parameter if the mean of the sampling distribution equals the parameter in question. (pick one )

1. No, it is biased. The mean of the distribution is p.2. No, it is biased. The mean of the distribution is . 3. Yes, it is unbiased. The mean of the distribution is . 4.Yes, it is unbiased. The mean of the distribution is .5.Yes, it is unbiased. The mean of the distribution is p.6.No, it is biased. The mean of the distribution is

14. Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a) Suppose n = 39 and p = 0.31. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.)

---Select--- Yes No , p  ---Select--- can cannot be approximated by a normal random variable because  ---Select--- nq does not exceed both np and nq exceed np and nq do not exceed np does not exceed np exceeds nq exceeds  .

What are the values of _p and _p? (Use 3 decimal places.)

(b) Suppose n = 25 and p = 0.15. Can we safely approximate p by a normal distribution? Why or why not?
---Select--- Yes No , p  ---Select--- cannot can be approximated by a normal random variable because  ---Select--- np and nq do not exceed both np and nq exceed nq does not exceed np exceeds np does not exceed nq exceeds  .

(c) Suppose n = 57 and p = 0.24. Can we approximate p by a normal distribution? Why? (Use 2 decimal places.)

---Select--- Yes No , p  ---Select--- can cannot be approximated by a normal random variable because  ---Select--- np exceeds both np and nq exceed nq exceeds np does not exceed nq does not exceed np and nq do not exceed  .

What are the values of _p and _p? (Use 3 decimal places.)

Solution

14.a) Suppose n = 39 and p = 0.31.If both np and np(1-p) are greater than 10, then normal approximation to binomial is suitable.

Here, np=39*0.31=12.09, np(1-p)=8.34

Thus, p bar can not be approximated by normal distribution.

b) Suppose n = 57 and p = 0.24.If both np and np(1-p) are greater than 10, then normal approximation to binomial is suitable.

Here, np=57*0.24=13.68, np(1-p)=10.39

Thus, p bar can be approximated by normal distribution.