Question

Kathy only receives surveys from 221 people despite expecting to receive 300 surveys. How will this affect the width and the margin of error of her confidence interval for a population mean? Assume that the population standard deviation is unknown and the population distribution is approximately normal.

Answer 1

Solution:

Consider the process of constructing the confidence interval for the population mean.

When the population standard deviation is unknown , then we use t distribution.

The margin of error is given by

E =  /2,d.f. * ( / n)

where s is the sample standard deviation and n is the sample size.

Since n is at denominator , so the margin of error is inversely proportional to the sample size.

As the sample size decreases , the margin of error and hence the width of the interval increases.

d.f. = n - 1

As sample size decreases , the d.f. also decreases.

For smaller value of d.f. , the larger is the  /2,d.f.

So , as sample size decreases , the value of  /2,d.f. increases and hence the margin of error also increases.

In the given example , the sample size is decreases from 300 to 221.

So , the width and the margin of error would be increased.