Answer to Question #328180 in Statistics and Probability for Lhiam

Question

A researcher is interested in estimating the average monthly salary of bank managers in NCR. He wants to be 90% confident that his estimate is correct. If the standard deviation is ₱8,500, how large a sample is needed to get the desired information and to be accurate within ₱1,150?

Answer 1

Let n - the sample size we want to find,

t - t-value for sample size n and confidence 90%,

$\sigma=8500,\space\Delta=1150\\ \Delta=t\frac{\sigma}{\sqrt{n}}\Rarr\frac{\sqrt{n}}{t}=\frac{\sigma}{\Delta}\approx7.3913$

Then we should look on t-values table on column for 90% confidence and find t and degrees of freedom (which is equal to n - 1), such that $\frac{\sqrt{n}}{t}=7.3913$

We can see from that table that $\frac{\sqrt{n}}{t}$ is growing up when n increases.

For n = 101 $\frac{\sqrt{n}}{t}=\frac{\sqrt{101}}{1.66}\approx6.05<7.3913$

When n increases further, t is almost not changing, so for estimation we can use t for 100 degrees of freedom, which is 1.66 (real t-value for n bigger than 101 is lower than 1.66, so real n is slightly lower than our estimate).

$n=(t\frac{\sigma}{\Delta})^2\approx(1.66\cdot7.3913)^2\approx150$

Answer: sample size should be equal or greater than 150.