The mean number of travel days per year for salespeople employed by hardware distributors needs to be estimated with a 0.80 degree of confidence. For a small pilot study the mean was 176 days and the standard deviation was 12 days. If the population mean is estimated within three days, how many salespeople should be sampled?
a. 613
b. 756
c. 2,450
d. 27
To determine the number of salespeople that should be sampled, we can use the formula for calculating the sample size required for estimating a population mean.
The formula is:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-value corresponding to the desired degree of confidence (0.80 in this case)
σ = standard deviation of the population
E = margin of error (the maximum difference between the sample mean and the population mean, which is given as 3 days in this case)
Let's plug in the given values:
Z = Z-value for a 0.80 degree of confidence is approximately 1.28 (you can refer to a standard normal distribution table or use a calculator)
σ = standard deviation = 12 days
E = margin of error = 3 days
n = (1.28 * 12 / 3)^2
n = (15.36 / 3)^2
n = 5.12^2
n ≈ 26.24
Since we cannot have a fraction of a salesperson, we need to round up to the nearest whole number. Therefore, the required sample size is 27 salespeople.
So, the answer is d. 27.