Computer Services wants to estimate the mean number of hours college students use on campus computers each day. They want to be within .1 hour of the true value with 99% confidence. From a previous study, the standard deviation is assumed to be .6 hour. What size sample is needed? Show any work necessary to support your answer
Formula:
n = [(z-value) * sd/E]^2
...where n = sample size, z-value will be 2.575 using a z-table to represent 99% confidence, sd = .6, E = .1, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
To determine the sample size needed, we can use the formula for calculating the sample size for estimating a population mean:
n = (Z * σ / E)^2,
where:
- n is the required sample size,
- Z is the Z-score corresponding to the desired level of confidence,
- σ is the standard deviation of the population, and
- E is the desired margin of error.
In this case, the desired level of confidence is 99% (which equates to a Z-score of 2.58, obtained from a standard normal distribution table for a 99% confidence level where the tail area is 0.005 on each side).
σ, the standard deviation, is given as 0.6 hours,
E, the desired margin of error, is 0.1 hours.
Substituting these values into the formula:
n = (2.58 * 0.6 / 0.1)^2
= (1.548)^2
= 2.393
Since sample sizes cannot be fractional, we always round up to the nearest whole number. Hence, the required sample size to estimate the mean number of hours college students use on campus computers each day with a 99% confidence and a margin of error of 0.1 hour is 3.
Therefore, Computer Services would need a sample size of at least 3 college students to estimate the mean number of hours used on campus computers each day.