Question

In a clinical​ study, 4500 healthy subjects aged​ 18-49 were vaccinated with a vaccine against a seasonal illness. Over a period of roughly 28​ weeks, 32 of these subjects developed the illness. Complete parts a through e below.

a. Find the point estimate of the population proportion that were vaccinated with the vaccine but still developed the illness.

The point estimate is?

​(Round to five decimal places as​ needed.)

b. Find the standard error of this estimate.

The standard error of this estimate is?

​(Round to five decimal places as​ needed.)

c. Find the margin of error for a 95​% confidence interval.

The margin of error is?

​(Round to five decimal places as​ needed.)

d. Construct the 95​% confidence interval for the population proportion. Interpret the interval.

The 95​% confidence interval for the population proportion is

​(Round to five decimal places as​ needed.)

Interpret the interval. Choose the correct answer below.

A.With 95​% ​confidence, the limits of the confidence interval contain the proportion of healthy people aged​ 18-49 who are vaccinated with the vaccine who do not develop the illness.

B.The proportion of all people who are vaccinated with the vaccine but still develop the illness falls within the limits of the confidence interval 95​% of the time.

C.With 95​% confidence, the limits of the confidence interval contain the proportion of healthy people aged​ 18-49 who are vaccinated with the vaccine but still develop the illness.

D.The proportion of healthy people aged​ 18-49 who are vaccinated with the vaccine but still develop the illness falls within the limits of the confidence interval 95​% of the time.

e. Is it safe to conclude that fewer than​ 1% of all the people aged​ 18-49 vaccinated with the vaccine will develop the​ illness? Explain by using the results from part d.

A. Yes​, it can be​ concluded, since the upper limit of the confidence interval is less than 0.01.

B. No​, it should not be​ concluded, since the upper limit of the confidence interval is greater than 0.01.

C. ​Yes, it can be​ concluded, since the lower limit of the confidence interval is less than 0.01.

D. ​No, it should not be​ concluded, since the lower limit of the confidence interval is greater than 0.01.

Answer 1

a) The point estimate of the population proportion that were vaccinated with the vaccine but still developed the illness is computed here as:

p = x/n = 32/4500 = 0.00711

Therefore 0.00711 is the required point estimate value here.

b) The standard error of estimate here is computed as:

Therefore 0.00125 is the required standard error here.

c) From standard normal tables, we have:
P( -1.96 < Z < 1.96) = 0.95

Therefore the margin of error here is computed as:

Therefore 0.00246 is the required margin of error here.

d) The confidence interval for the proportion here is computed as:

This is the required proportion confidence interval here.