sample mean is 68.
population standard deviation is 3.
90% confidence level requires the critical z-score to be plus or minus 1.644853626 according to my TI-84 Plus calculator.
this is found by taking the alpha for 90% confidence level and dividing it by two to get alpha / 2 that will be on each side of the confidence interval.
90% confidence level gets you an alpha of 100% - 90% = 10% / 2 which gets you an alpha / 2 of 5%.
this tells you that the area under the normal distribution curve to the left of your lower confidence interval limit is 5% or .05 and it tells you that the area under the normal distribution to the right of your higher confidence interval limit is 5% or .05 as well.
note that an area under the normal distribution curve to the right of your higher confidence interval limit of 5% means that the area under the normal distribution curve to the left of your higher confidence interval limit is 100% - 5% = 95%, or 1 - .05 = .95.
the z-scores associated with the lower limit and the higher limit are -1.644853626 and 1.644853626 respectively.
this can be seen in the following graph.
those results are rounded to 3 decimal places, which is usually enough accuracy for practical purposes.
the calculator used to show the graph can be found at http://davidmlane.com/hyperstat/z_table.html
up to this point, you know that the critical z-scores for a 90% confidence level are plus or minus 1.645 (we'll use the rounded version from here on).
the formula for z-score is z = (x - m) / s
in the context of this problem:
z = the critical z-score
m = the mean of the sample
x = the raw score of the sample
s = the standard error of the distribution of sample means.
the formula for s is s = standard deviation of the population divided by the square root of the sample size.
this becomes s = 3 / sqrt(36) = 3 / 6 = .5
the formula for the z-score becomes z = (x - m) / .5
when z = 1.645 and m = 68, the formula becomes 1.645 = (x - 68) / .5
solve for x to get x = 1.645 * .5 + 68 = 68.8225
when z = -1.645 and m = 68, the formula becomes -1.645 = (x - 68) / .5
solve for x to get x = -1.645 * .5 + 68 = 67.1775
this tells you that, if you take an infinite number of samples of size 36, you will find that the sample mean will be somewhere between 67.1775 and 68.8225 for approximately 90% of the samples taken.
here's the graph that shows you that this is pretty accurate.
