Question

An epidemiologist is worried about the prevalence of the flu in East Vancouver and the potential shortage of vaccines for the area. She will need to provide a recommendation for how to allocate the vaccines appropriately across the city. She takes a simple random sample of 333 people living in East Vancouver and finds that 36 have recently had the flu.

For each of the following statements, specify whether the statement is a correct interpretation of the 95% confidence interval for the true proportion of East Vancouver residents who have recently had the flu.

A. 10.81% (36/333) of East Vancouver residents have recently had the flu. true/ false

B. There is a 95% probability that the true proportion of East Vancouver residents who have recently had the flu equals 36/333. true/false

C. If another random sample of 333 East Vancouver residents is drawn, there is a 95% probability that the sample proportion of East Vancouver residents who have recently had the flu equals 36/333. true/false

D. If many random samples of 333 East Vancouver residents are drawn, 95% of the resulting confidence intervals will contain the value of the true proportion of East Vancouver residents who have recently had the flu. true/false

E. If many random samples of 333 East Vancouver residents are drawn, 95% of the resulting confidence intervals will contain the value 36/333. true/false

Part 2

An epidemiologist is worried about the prevalence of the flu in East Vancouver and the potential shortage of vaccines for the area. She will need to provide a recommendation for how to allocate the vaccines appropriately across the city. She takes a simple random sample of 340 people living in East Vancouver and finds that 33 have recently had the flu.

Suppose that the epidemiologist wants to re-estimate the population proportion and wishes for her 95% confidence interval to have a margin of error no larger than 0.04. How large a sample should she take to achieve this? Please carry answers to at least six decimal places in intermediate steps.

Sample Size =

Answer 1

PART 1)

a) False. 36 out of 333 people in the sample were found to have caught the flue recently. This is essentially related to the sample, not to the population. We associate a certain level of confidence that claims with %age surety about the claim made towards the population. The true value might not be estimated with the sample, and therefore, associating directly with the population parameters, the estimation of sample mean is absurd.

b) False. With respect to confidence interval, we do known one thing that point estimation is a different concept altogether. Therefore, interval estimation puts forward limits within which the true value of the population parameter would lie. Had it been point estimation, we would have said that there is 95% surety (confidence) that the true value would be equal to 10.81%.

c) False, again the statement mentions about point estimation, whereas our view point is directly related to confidence interval estimation.

d) True( in very general sense, false with respect to the question), This is a very general statement about interval estimation. Given that

Pr[ { (t - E(t))/(SD) } < Z ] = 1- %

and hence, E(t) lies in the interval given as { t - SD*Z , t+SD*Z}

This essentially means that if many samples are drawn, 95% of those will contain the true value in the interval. Note that the true value will be affected by what is offered as t (estimator) under Ho, which needs to be tested.

e) True, again this is true because out of many samples drawn for the same hypothesis, it is to be concluded that 95% of those will contain the true value taken to be as 10.81% under Ho. Relate with the formula provided above, which naturally clears off the doubt in this regard.

Part 2) please look into the pic attached.

The sample size should be 147.