Assume that previous fast-food research has shown that 80 percent of the consumers like curly French fries. The researcher wishes to have a standard error 6 percent or less and be 95 percent confident of an estimate to be made about curly French fry consumption from a survey. What sample size should be used for a simple random sample?
rando
To determine the sample size for a simple random sample, we can use the formula:
n = (Z * σ / MOE)²
where:
n = sample size
Z = Z-score corresponding to the desired confidence level
σ = standard deviation
MOE = margin of error
In this case, the researcher wants to be 95% confident and have a standard error of 6%. The Z-score corresponding to a 95% confidence level is approximately 1.96.
Given that the previous research showed that 80% of consumers like curly French fries, we can assume that the standard deviation is:
σ = √(p * (1 - p) / n)
where:
p = proportion of consumers who like curly French fries
n = sample size
Let's plug in the values and solve for n:
σ = √(0.80 * (1 - 0.80) / n)
To have a margin of error of 6%, we can set the standard deviation equal to MOE:
6% = √(0.80 * (1 - 0.80) / n)
Rearranging the equation, we have:
n = 0.80 * (1 - 0.80) / (0.06)²
n = 0.16 / (0.06)²
n = 44.44
Since you can't have a fraction of a person, we need to round up to the nearest whole number.
n = 45
Therefore, a sample size of 45 should be used for a simple random sample in order to estimate curly French fry consumption with a standard error of 6% or less and be 95% confident.
To determine the sample size needed for a simple random sample, we can use the formula for the sample size required to estimate a population proportion:
n = (z^2 * p * (1-p)) / (E^2)
Where:
n = Sample size
z = Z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to a z-score of approximately 1.96)
p = Estimated proportion (in this case, the proportion of consumers who like curly French fries, which is 0.8)
E = Maximum tolerable error (in this case, 6% or 0.06)
Substituting the values into the formula:
n = (1.96^2 * 0.8 * (1-0.8)) / (0.06^2)
Simplifying the equation:
n = (3.8416 * 0.16) / 0.0036
n ≈ 170.1111
Since we cannot have a fraction of a person in a sample, we need to round up to the nearest whole number:
n = 171
Therefore, to ensure a standard error of 6% or less and a 95% confidence level, a minimum sample size of 171 participants should be used for a simple random sample.