# Normal Distribution and Random Sample

**Topics:**Normal distribution, Arithmetic mean, Sample size

**Pages:**6 (1373 words)

**Published:**July 2, 2013

1.What is the difference between [pic] and[pic]? Between s and[pic]? (10 points)

2.Explain the difference between [pic] and [pic] and between [pic] and[pic]? (10 points)

3.Suppose that a random sample of size 64 is to be selected from a population having [pic] and standard deviation 5.

(a)What are the mean and standard deviation of the [pic] sampling distribution? Can we say that the shape of the distribution is approximately normal? Why or why not? (10 points)

(b)What is the probability that [pic] will be within 0.5 of the population mean? (5 points)

(c)What is the probability that [pic] will differ from the population mean by more than 0.7? (5 points)

4.In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. Furthermore, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and staff on campus is 150 pounds with a standard deviation of 27 pounds, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is taken:

(a)What is the mean and standard deviation of the [pic] = sample mean distribution? Can we assume the [pic] distribution is normal? Explain. (10 points)

(b)What average weights of for a sample of 16 people will result in the total weight exceeding the weight limit of 2500 pounds? (5 points)

(c)What is the chance that a random sample of 16 persons on the elevator will exceed the weight limit? (5 points)

5.A random sample is to be selected from a population that has a proportion of successes[pic]. Determine the mean and standard deviation of the sampling distribution of p for each of the following sample sizes. (5 points each)

(a)n = 10

(b)n = 20

(c)n = 29

(d)n = 200

6.For which of the sample sizes above in #5 would the sampling distribution of p be approximately normal when[pic]? (10 points)

7.The article “Should Pregnant Women Move? Linking Risks for Birth Defects with Proximity to Toxic Waste Sites” (Chance [1992]) reported that in a large study carried out in the state of New York, approximately 30% of the study subjects lived within one mile of a hazardous waste site. Let [pic] denote the proportion of all New York residents who live within 1 mile of such a site, and suppose that[pic]. (5 points each)

(a)Would p based on a random sample of only 10 residents have approximately a normal distribution? Explain why or why not.

(b)What are the mean and standard deviation of p based on a random sample of size 400?

(c)When n = 400, what is[pic]?

8.A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected, and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows: (5 points each)

1031561188912514712210913899

(a)Let [pic] denote the average gas usage during January by all houses in this area. Compute an unbiased point estimate of [pic].

(b)Let [pic]denote the proportion of houses in this area that use more than 100 therms during January. Compute a point estimate of [pic].

9.The article “Doctors Cite Burnout in Mistakes” (San Luis Obispo Tribune, March 5, 2002) reported that many doctors who are completing their residency have financial struggles that could interfere with training. In a sample of 115 residents, 38 reported that they worked moonlighting jobs and 22 reported a credit card debt of more than $3000. Suppose that it is reasonable to consider this sample as a random sample of all medical residents in the United States.

(a)Construct and interpret a 95% confidence...

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