In a random sample of 75 individuals, it was found that 52 of them prefer coffee to tea. What is the margin of error for the true proportion of all individuals who prefer coffee?
Since we're speaking of proportion in this problem, the margin of error for the true proportion of all individuals who prefer coffee is E = (z critical value)*Sqrt(p-hat)*(1-p-hat)/n), where p-hat is the sample proportion x/n (=52/75 in this case) and n is the sample size (=75 in this case). The z-critical value depends on the level of confidence desired. For 95% confidence, that z-critical value would be 1.96. If the level of confidence is not specified, assuming that it's 95% is reasonable. A greater level of confidence would link to a larger z critical value and thus a greater margin of error, and vice versa.
Assuming that the level of confidence here is 95%, E = margin of error = 1.96*Sqrt[(52/75)(1-(52/75)/75).
To determine the margin of error for the true proportion of all individuals who prefer coffee, we need to use the formula for the margin of error in a proportion:
Margin of Error = Z * sqrt((p * (1 - p)) / n)
Where:
- Z is the z-score corresponding to the desired level of confidence
- p is the sample proportion
- n is the sample size
In your case, the sample size is 75 individuals, and the proportion who prefer coffee is 52 out of 75. First, we need to find the sample proportion, which is calculated by dividing the number of individuals who prefer coffee by the total sample size:
p = 52/75 = 0.6933
Next, we need to determine the z-score corresponding to the desired level of confidence. Let's assume a 95% confidence level, which corresponds to a z-score of 1.96.
Z = 1.96
Now, we can substitute these values into the formula to calculate the margin of error:
Margin of Error = 1.96 * sqrt((0.6933 * (1 - 0.6933)) / 75)
Using a calculator, the calculation is as follows:
Margin of Error = 1.96 * sqrt((0.6933 * 0.3067) / 75) = 0.0754
Therefore, the margin of error for the true proportion of all individuals who prefer coffee is approximately 0.0754, which means we can be 95% confident that the true proportion falls within 0.0754 units of the sample proportion.