It is known that the standard deviation for the number of minutes women spend applying makeup everyday is 17 minutes if you want to construct a 95% confidence interval for the mean amount of time women spend applying make up with a margin of error of no more than 5 mins. At least how many women must you survey
95% = mean ± 1.96SEm
1.96 SEm ≤ 5
SEm = SD/√n
I'll let you do the calculations.
To determine the minimum sample size required, we need to consider the formula for the sample size calculation of a confidence interval:
n = (Z * σ / E) ^ 2
Where:
- n is the sample size
- Z is the Z-score corresponding to the desired level of confidence (in this case, for a 95% confidence level, Z ≈ 1.96)
- σ is the standard deviation of the population (17 minutes in this case)
- E is the maximum margin of error allowed (5 minutes in this case)
Substituting the given values into the formula:
n = (1.96 * 17 / 5) ^ 2
n ≈ (33.32 / 5) ^ 2
n ≈ 6.664 ^ 2
n ≈ 44.351296
Since the sample size must be a whole number, we round up to the nearest whole number:
n ≈ 45
Therefore, at least 45 women must be surveyed to construct a 95% confidence interval for the mean amount of time women spend applying makeup, with a margin of error no more than 5 minutes.