Question

A food store owner wants to determine the type of products customers prefer to buy. The owner surveyed 40 customers on a day when the store had 335 customers, and found that 40% of those surveyed preferred to buy organic products over non-organic products. Assuming a 95% confidence level, which of the following statements holds true?
a) As the sample size is appropriately large, the margin of error is ±0.127.
b) As the sample size is appropriately large, the margin of error is ±0.152.
c) As the sample size is too small, the margin of error is ±0.127.
d) As the sample size is too small, the margin of error cannot be trusted.

Answer 1

There is not a fixed boundary line as to when the sample size should be considered appropriately large but the rule of thumb is a sample size larger than 30 is sufficient enough to assume that the data is normally distributed. Since the sample size in this case is 40, we can consider it appropriately large and use Normal Distribution to deduce the results.

Sample Size = n = 40
Proportion of customer who prefer organic food= p = 40% = 0.40
Confidence Level = 95%
Z score at the confidence Level = z = 1.96

Margin of Error for the population proportion is calculated as:

$M.E=+-z \sqrt{ \frac{p(1-p)}{n} }$

Using the values, we get:

$M.E=+-1.96 \sqrt{ \frac{0.4*0.6}{40} }=+-0.152$

Thus, the option B gives the correct answer.
As the sample size is appropriately large, the margin of error is ±0.152.