Question

A researcher wishes to estimate, with 95% confidence, the population proportion of adults who eat fast food four to six times per week. Her estimate must be accurate within 5% of the population proportion.

(a) What is the minimum sample size needed assuming that no prior information is available?
(b) What is the minimum sample size needed using a prior study that found that 20% of the respondents said they eat fast food four to six times per week?
(c) How do the results from (a) and (b) compare?

Options:
A) Having an estimate of the population proportion reduces the minimum sample size needed
B) Having an estimate of the population proportion has no effect on the minimum sample size needed
C) Having an estimate of the population proportion raises the minimum sample size needed

Answer 1

For establishing the minimum sample size needed for estimating a population proportion with 95% confidence and a 5% margin of error, without prior knowledge, we round up the calculated sample size of 384.16 to 385. With an estimated population proportion of 0.2 from a prior study, the minimum required sample size calculated is smaller, rounded up to 246.

Step-by-step explanation:

To answer this question, we calculate the minimum sample size necessary for estimating a population proportion with a given level of confidence and margin of error using the formula for sample size in estimating proportions:

n = (Z² * p * (1-p)) / E²

Where:

• Z is the Z-score associated with the desired level of confidence
• p is the estimated population proportion (or 0.5 if no prior estimate is available)
• E is the desired margin of error

(a) Without prior information, we use p = 0.5 for maximum variability. With a 95% confidence level, the Z-score is approximately 1.96. The desired margin of error (E) is 0.05. Plugging these values into the formula, we get:

n = (1.96² * 0.5 * (1-0.5)) / 0.05²

n = 384.16

Therefore, a minimum sample size of 385 is needed (since we always round up to the next whole number in sample size calculations).

(b) Using a prior study with p = 0.2, we get:

n = (1.96² * 0.2 * (1-0.2)) / 0.05²

n = 245.86

So, a minimum sample size of 246 is needed.

(c) Comparing the results, we can conclude that:

Having an estimate of the population proportion reduces the minimum sample size needed.