An estimate of the time required to perform a certain job is needed. The study yielded a mean of 6.4 minutes and a standard deviation of 2.1 minutes. The desired confidence is 0.95. How many observations will be needed if the desired maximum error is:
a. +or – 10% of the sample mean
b. ½ minute
To determine the number of observations needed to estimate the time required to perform a certain job with a desired maximum error, we can use the formula for sample size estimation.
The formula for sample size estimation is:
n = (Z * σ / E)^2
Where:
n = required number of observations
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
σ = standard deviation of the population
E = maximum error (also known as the margin of error)
Let's calculate the number of observations needed for each case:
a. + or – 10% of the sample mean:
For this case, the maximum error (E) will be equal to 0.10 times the sample mean (6.4 minutes) since we want the maximum error to be within 10% of the sample mean.
E = 0.10 * 6.4 = 0.64 minutes
Substituting the values into the formula:
n = (1.96 * 2.1 / 0.64)^2
n ≈ 22.46
Since we cannot have a fraction of observations, we would need at least 23 observations (rounded up) to estimate the time required to perform the job with a maximum error of + or – 10% of the sample mean.
b. ½ minute:
For this case, the maximum error (E) is given directly as 0.5 minutes.
Substituting the values into the formula:
n = (1.96 * 2.1 / 0.5)^2
n ≈ 66.30
In this case, since we cannot have a fraction of observations, we would need at least 67 observations (rounded up) to estimate the time required to perform the job with a maximum error of ½ minute.