The Grocery Manufacturers reported that 68% of consumers read the ingredients listed on a product’s label. Assume the population proportion is 𝑝 = 0.68 and a sample of 450 consumers is selected from the population of 1000. What sample size will give at least a 0.95 probability that the sample proportion will be within ±0.04 of the population proportion?
To determine the sample size that will give at least a 0.95 probability that the sample proportion will be within ±0.04 of the population proportion, we can use the formula for sample size calculation for proportions.
The formula for sample size (n) for proportions is:
n = (Z^2 * p * (1 - p)) / E^2
where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (in this case, for a 0.95 probability, Z = 1.96)
- p is the estimated population proportion
- E is the maximum margin of error allowed
In this case, the estimated population proportion is p = 0.68, and the maximum margin of error allowed is E = 0.04.
Substituting these values into the formula:
n = (1.96^2 * 0.68 * (1 - 0.68)) / 0.04^2
Calculating this expression:
n ≈ (3.8416 * 0.68 * 0.32) / 0.0016 ≈ 62.5024 / 0.0016 ≈ 39063.9
Rounding up to the nearest whole number, we get the sample size:
n ≈ 39064
Therefore, a sample size of at least 39,064 consumers needs to be selected in order to have at least a 0.95 probability that the sample proportion will be within ±0.04 of the population proportion.