Consider the following data set:

Question

Consider the following data set:

8 7 13 3 6

4 8 6 3 4

0 1 11 7 1

8 6 12 13 10

9 11 3 2 9

7 1 16 3 2

10 15 2 5 8

8 17 2 5 5

(a) Calculate ¯x and s

(b) Calculate the percentages of observations (numbers in the data set)

that fall in the intervals ¯x ± s, ¯x ± 2s, and ¯x ± 3s

Answer 1

if X = {8,7,13,3,6,4,8,6,3,4,0,1,11,7,1,8, 6,12 ,13,10,9,11,3,2,9,7,1, 6,3,2,10,15,2,5,8,8,17,2,5,5}

then n = 40

Xbar =(1/40) ∑X_{i for all i=1:n} which is 6.675

s = √(1/(n-1) * ∑( x_i-xbar)² or 4.40563

Frequencies of numbers is

0 1 2 3 4 5 6 7 8 9 10 11 12

0.025 0.075 0.100 0.100 0.050 0.075 0.075 0.075 0.125 0.050 0.050 0.050 0.025

13 15 16 17

0.050 0.025 0.025 0.025

Counts of numbers is

0 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17

1 3 4 4 2 3 3 3 5 2 2 2 1 2 1 1 1

_{note: using standard deviations to create intervals assumes normal distributions which is not true in this case.}

First interval is 2.269 to 11.081 - which has a total of 26 numbers in it or 65%

Second interval is -2.269 to 15.487 - which has a total of 38 numbers in it or 95%

Third interval is -6.542 to 19.893- which has a total of 40 numbers in it or 100%