The proportion of all consumers favoring a new product might be a slow as 0.20 or as high as 0.60.A random sample is to be used to estimate the proportion of the consumers who favor the the new product to with in plus or minus 0.05,with a confidence coefficient of 90%. To be on the safe, what is the sample size shuold be used.
We can use the formula for sample size:
n = [(z-score)^2 * p * (1-p)] / E^2
where:
z-score = 1.645 (from a Z-table for 90% confidence level)
p = 0.5 (a conservative estimate assuming no prior knowledge about the true proportion)
E = 0.05 (the margin of error)
Plugging in the values, we get:
n = [(1.645)^2 * 0.5 * (1-0.5)] / (0.05)^2
n = 676.73
Rounding up to the nearest whole number, we get a sample size of 677.
Therefore, a sample size of at least 677 consumers should be used to estimate the proportion of those who favor the new product within plus or minus 0.05, with a 90% confidence level.
To determine the sample size needed in order to estimate the proportion of consumers who favor the new product within plus or minus 0.05, with a confidence coefficient of 90%, we can use the following formula for calculating the sample size for proportions:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (90% confidence level corresponds to Z = 1.645)
p = estimated proportion (we can use the maximum proportion of 0.60 as a conservative estimate)
E = margin of error (in this case, 0.05)
Plugging in the values, we get:
n = (1.645^2 * 0.60 * (1 - 0.60)) / 0.05^2
Simplifying the equation:
n = (2.706 * 0.24) / 0.0025
n = 259.44 / 0.0025
n ≈ 103776
Therefore, to be on the safe side, a sample size of at least 103776 should be used.