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.