Question

A NEW STUDY IS DESIGNED TO ESTIMATE THE AVERAGE GROCERY SPENDING. INDIVIDUAL RECORDS ARE ASSUMED TO BE INDEPENDENT NORMALLY DISTRIBUTED WITH THE POPULATION STANDARD DEVIATION = $32. RESEARCHERS WANT TO HAVE THE MARGIN OF ERROR OF THE 95% CONFIDENCE INTERVAL NO BIGGER THAN $15. WHAT IS THE SMALLEST SAMPLE SIZE NEEDED? SHOW THE FOLLOWING ELEMENTS OF YOUR DECISION.

CRITICAL VALUE
SMALLEST SAMPLE SIZE
IS THE SAMPLE SIZE OF 25 SUFFICIENT FOR THE GOALS OF THE STUDY?

Answer 1

CRITICAL VALUE

1.96

SMALLEST SAMPLE SIZE

18

IS THE SAMPLE SIZE OF 25 SUFFICIENT FOR THE GOALS OF THE STUDY?

Yes, it is.

Explanation:

The sample size n in Simple Random Sampling is given by

`\bf n=(z^2s^2)/(e^2)`

where

z = 1.96 is the critical value for a 95% confidence level

s = 32 is the population standard deviation

e = 15 is the margin of error

so

`\bf n=(z^2s^2)/(e^2)=((1.96)^2(32)^2)/((15)^2)=17.48\approx 18`

CRITICAL VALUE

1.96

SMALLEST SAMPLE SIZE

18

IS THE SAMPLE SIZE OF 25 SUFFICIENT FOR THE GOALS OF THE STUDY?

Yes, it is.