Question

A random sample of 16 pharmacy customers showed the waiting times below (in minutes). Find a 90 percent confidence interval for μ, assuming that the sample is from a normal population.
21 22 22 17 21 17 23 20
20 24 9 22 16 21 22 21

Answer 1

values are shown below: ----- 21 22 22 17 21 17 23 20 20 24 9 22 16 21 22 21 ----- sum and average of all values is shown below: ----- 318 = sum of all numbers 16 = number of occurrences 19.875 = sum of all numbers divided by number of occurrences = average = mean = {{{mu}}} ----- square of all values minus the mean is shown below: example: 1.265625 = {{{(21-19.875)^2}}} = {{{(x - mu)^2}}} ----- 1.265625 4.515625 4.515625 8.265625 1.265625 8.265625 9.765625 0.015625 0.015625 17.015625 118.265625 4.515625 15.015625 1.265625 4.515625 1.265625 ----- 199.75 = sum of all deviations squared. 16 = number of occurrences 3.533323506 = standard deviation of population = {{{sigma = sqrt(sum((x-mu)^2)/n))}}} ----- to find the 90% confidence interval for the population (2 tailed confidence interval assumed), do the following: ----- go to the following website: http://davidmlane.com/hyperstat/z_table.html ----- got to the bottom graph. enter a mean of 19.875 enter a sd of 3.5333 enter a shaded area of .9 select between ----- your answer is that at a 90% confidence level, the waiting times can be between 14.0632 minutes and 25.6868 minutes. ----- to see how this is represented on a z score, do the following: go to the bottom graph. enter a mean of 0 enter a sd of 1 enter a shaded area of.9 select between ----- your answer is that at a 90% confidence level, the waiting times can be between 1.6449 standard deviations below the mean and 1.6449 standard deviations above the mean. ----- 1.6449 standard deviations is 1.6449 * the standard deviation of 3.5333 = 5.8119 ----- the mean of 19.875 minus 5.8119 = 14.063 which is close enough to 14.0632 to have the discrepancy be explained by rounding error. similarly, the mean of 19.875 plus 5.8119 = 25.6869 is close enough to also have the discrepancy be explained by rounding error. ----- answer to your question: mean = 19.875 standard deviation equal 3.5333 at 90% confidence interval waiting can be between 14.0632 and 25.6869 minutes this is between minus 1.6449 and plus 1.6449 standard deviations from the mean. -----