James B. answered • 03/09/23
B.S. in Math with 2+ years tutoring experience
we begin by calculating the mean which is (86 + 83 + 86 + 95 + 75 + 94 + 116 + 100 + 85)/9 = 81.67 rounded to two decimal places. Since this is a sample its standard deviation can be calculated by (sum i = 1 to n (x_i - mean x)^2/(n-1))^(1/2) where each x is an observation in the sample data and n is the number of observations.
86 - 81.67 = 4.33
83 - 81.67 = 1.33
86 - 81.67 = 4.33
95 - 81.67 = 13.33
75 - 81.67 = -6.67
94 - 81.67 = 12.33
116 - 81.67 = 34.33
100 - 81.67 = 18.33
85 - 81.67 = 3.33
squaring each of these results and summing them together gives us 18.75 + 1.78 + 18.75 + 177.69 + 44.49 + 152.03 + 1178.55 + 335.99 + 11.09 = 1603.13
1603.13/(n-1) = 1603/8 = 200.39
the square root of 200.39 = 14.16 and thus we have a standard deviation 14.16 with a mean 81.67. Great
Now we want a 90% confidence interval, we can obtain this by getting the degrees of freedom which is the number of values that are free to vary which is just n - 1 (all values can vary but one), or 8. Using a t-table, which you can look up online, we get 1.860 which is our confidence level value as referred to in the confidence interval formula. Next we take our sample standard deviation, 14.16 and divide it by the square root of the total observations, which is (9)^(1/2) = 3. Combining this all together we have
Confidence interval = 81.67 +/- 1.860*(14.16/3) = 81.67 +/- 8.7792
Upper limit = 90.45
Lower limit = 72.89
