Question 1184081
since the standard deviation is taken from the sample, rather than from the population, use the t-score rather than the z-score.


sample size = 25
sample mean = 14.5%
sample standard deviation = 3.4%


standard error = sample standard deviation divided by square root of sample size = 3.4/sqrt(25) = 3.4/5 = .68


critical t-score with 24 degrees of freedom at 90% two-tailed confidence level = plus or minus 1.711.


critical t-score with 24 degrees of freedom at 95% two-tailed confidence level = plus or minus 2.064.


these were taken from the following table:


<a href = "http://www.math.odu.edu/stat130/t-tables.pdf" target = "_blank">http://www.math.odu.edu/stat130/t-tables.pdf</a>


they were also verified through use of the ti-84 plus scientific calculator.


with a critical t-score of plus or minus 1.711, the critical raw score is calculated as shown below:


-1.711 = (x - 14.5) / .68
solve for x to get x = -1.711 * .68 + 14.5 = 13.33652%


1.711 = (x - 14.5) / .68
solve for x to get x = 1.711 * .68 + 14.5 = 15.66345%


with a critical t-score of plus or minus 2.064, the critical raw score is calculated as shown below:


-2.064 = (x - 14.5) / .68
solve for x to get x = -2.064 * .68 + 14.5 = 13.09648%


2.064 = (x - 14.5) / .68
solve for x to get x = 2.064 * .68 + 14.5 = 15.90352%


at 90% confidence level, the range is from 13.33652% to 15.66345%
at 95% confidence level, the range is from 13.09648% to 15.90352%


the range is larger at 95% confidence level than at 90% confidence level.


this is to be expected because there needs to be less change of error at 95% confidence level than at 90% confidence level.


at 90% confidence level, 90% of the sample means, each with a sample size of 25, are expected to be within the range specified.


at 95% confidence level, 95% of the sample means, each with a sample size of 25, are expected to be within the range specified.