Question 1173384
sample size is 49
mean of sample is 131.8
standard deviation of sample is 6.25


you want to set up a 90% confidence interval.


degrees of freedom is 49 - 1 = 48.


standard error = standard deviation of sample divided by square root of sample size.


you will be using t-scores with 48 degrees of freedom.


standard error = 6.25 / sqrt(49) = 6.25 / 7 = .8928571429.
i rounded this to .892857 which is probability a little more than close enouogh.


in the t-score table, with 90% confidence interval, one tailed alpha would be 5% and two tailed alpha would be 10% (5% on each end).


here's the table i used.


<a href = "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf" target = "_blank">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>


that table only showed degrees of freedom of 40 and 60.
48 degrees of freedom is somewhere in between.


fortunately, there is a t-score calculator online that can be used to get more accurate critical t-score.


that t-score calculator can be found at <a href = "https://stattrek.com/online-calculator/t-distribution.aspx" target = "_blank">https://stattrek.com/online-calculator/t-distribution.aspx</a>


the calculator gets you the area to the left of the t-score.
to find the two tailed 10% conidence interval, i entered .05 in the p(T < t) box and a mean of 0 and a standard deviatiion of 1 and degrees of freedom of 48 and clicked on calculate.


calculator said the t-score was -1.677
since the normal distribution table is symmetric, my confidence interval would be between a t-score of -1.677 and a t-score of 1.677
that would be 5% area to the left of -1.677 and 5% area to the right of -1.677.


in the table, two tailed of 10% with 40 degrees of freedom gave a t-score of -1.684 and 60 degrees of freedom gave a t-score of 1.671.
about halfway between those t-score would yield a t-score of -1.6775 which is pretty close to -1.677 found by the calculator.


i used a critical t-score of plus or minus 1.677 for my 10% two tailed confidence interval.


once this is done, you need to use the t-score formula to find the raw score.


t-score formula is t = (x - m) / s
t is the t-score
x is the raw score
m is the mean
s is the standard error.


for t = -1.677, i got:
-1.677 = (x - 131.8) / .892857 
solve for x to get:
x = -1.677 * .892857 + 131.8 = 130.3026788


for t = 1.677, i got:
1.677 = (x - 131.8) / .892857
solve for x to get:
x = 1.677 * .892857 + 131.8 = 133.2973212


your answer should be:


90% confidence interval is between 130.3026788 and 133.2973212.