Question

PART I

A survey with an SRS of 2500 faculty members at US universities found that on average they had published 1.7 papers the previous 2 years with anSD of 2.3 papers. Find a 95% confidence interval for the average number of papers published by faculty members at US institutions in the previous2 years.

PART II

In 2001, a survey was carried with a simple random sample of 500 house-holds from 25,000 households in a certain town.

(a)239 of the households surveyed had internet access. If possible, find a 95% confidence interval for the percentage of all 25,000 households that have internet access. If this is not possible, explain why.

(b)7 of the households surveyed had 3 or more large screen TVs. If possible, find a 95% confidence interval for the percentage of all 25,000households that have 3 or more large screen TVs. If this is not possible, explain why.78

(c)121 had no car, 172 had one car and 207 had two or more cars.Estimate the percentage of 25,000 households with one or more cars and give a standard error for your estimate. If this is not possible, explain why.

PART III

Biologists often use a sampling technique calledCapture–Recapture to estimate sizes of populations, such as deer or fish, for which a complete census is impossible. For example, to count the deer population in a certain region, a biologist first captures, say, 50 deer, tags each, and releases them. Several weeks later, she captures 125 deer and finds that 12 of them were tagged. Let N= population size,M= size of first sample, n= size of second sample andR= number tagged in second sample.

(a)One way to approximate N is to equate the population proportion of tagged deer p=MN to the sample proportion of tagged deer bp=Rn.Use this to estimate the number of deer in the hypothetical example.

(b)Lots of assumptions are being made here (e.g tags don’t fall off!). Make a list of things that could go wrong and how they could affect the estimate for N.

(c)The estimate is based on a single sample, and we know there is always variation in random sampling. Construct a 95% confidence interval for the number of deer. (Start by constructing a 95% confidence interval for p).

Answer 1

dear student we can provide you with the solution of one question and 4 sub-question at a time.

Part I ) Since the sample is random and the sample size is large enough we will use Z distribution to calculate the confidence interval of mean.

The 95% confidence interval for the average number of papers published by faculty members at US institutions in the previous2 years is

Part II)

a) first we will calculate the 95% confidence interval of proportion then multiply the answer with 100 to get the percentage

95% confidence interval for the percentage of all 25,000 households that have internet access is (43.4% , 52.2%)

b) to calculate the confidence interval of proportion the number of successes and the number of failures must be greater or equal to 10.

Here the number of success is the number of households that have 3 or more large-screen TVs, which is 7 and less than 10 hence calculating the confidence interval of proportion is not possible and so does the a 95% confidence interval for the percentage of all 25,000 households that have 3 or more large screen TVs.

c) The number of households with one or more cars = 172 +207 = 379

the proportion of households with one or more cars is

in percentage, it is 75.8%

standard error for the estimate is

= 1.92%