Answer:
Explanation:
To determine the minimum sample size needed to have a 4 percentage point error for the proportion of orders that were carry-out, we can use the following formula:
n = (Z^2 * p * q) / E^2
where:
n is the sample size we want to determine
Z is the z-score associated with the desired level of confidence (let's assume a 95% confidence level, so Z = 1.96)
p is the estimated proportion of orders that were carry-out (we don't know this yet, so we'll use 0.5 as a conservative estimate)
q is 1 - p (i.e., the estimated proportion of orders that were delivery)
E is the desired margin of error (in this case, 0.04)
Using the given information, we can estimate that the proportion of orders that were carry-out is:
p = 1 - 431/778 = 0.446
Then, plugging in the values into the formula, we get:
n = (1.96^2 * 0.446 * 0.554) / 0.04^2 ≈ 602.25
Rounding up to the nearest whole number, the minimum sample size needed is 603.
Note that I used the formula for sample size calculation based on a finite population, since the total number of orders is known (778). If we used the formula for an infinite population, the result would be very similar (around 604). I calculated this by hand, without using any calculator program, but you could use a calculator or a spreadsheet to perform the calculations more efficiently.