Question

Consider a sample with 10 observations of 5, 11, 10, 6, 6, 15, 2, 15, 3, and 2. Use z-scores to determine if there are any outliers in the data; assume a bell-shaped distribution. (Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)

Answer 1

To determine if there are any outliers in the data, we need to calculate the z-scores for each observation.

The formula to calculate the z-score is:

z = (x - mean) / standard deviation

First, we need to calculate the mean:

mean = (5 + 11 + 10 + 6 + 6 + 15 + 2 + 15 + 3 + 2) / 10 = 7.5

Next, we need to calculate the standard deviation:

Step 1: Calculate the deviation for each observation:
(5 - 7.5) = -2.5
(11 - 7.5) = 3.5
(10 - 7.5) = 2.5
(6 - 7.5) = -1.5
(6 - 7.5) = -1.5
(15 - 7.5) = 7.5
(2 - 7.5) = -5.5
(15 - 7.5) = 7.5
(3 - 7.5) = -4.5
(2 - 7.5) = -5.5

Step 2: Square each deviation:
(-2.5)^2 = 6.25
(3.5)^2 = 12.25
(2.5)^2 = 6.25
(-1.5)^2 = 2.25
(-1.5)^2 = 2.25
(7.5)^2 = 56.25
(-5.5)^2 = 30.25
(7.5)^2 = 56.25
(-4.5)^2 = 20.25
(-5.5)^2 = 30.25

Step 3: Calculate the average of the squared deviations:
(6.25 + 12.25 + 6.25 + 2.25 + 2.25 + 56.25 + 30.25 + 56.25 + 20.25 + 30.25) / 10 = 20.5

Step 4: Take the square root of the average squared deviations:
sqrt(20.5) ≈ 4.53

Now we can calculate the z-scores for each observation:

z1 = (5 - 7.5) / 4.53 ≈ -0.55
z2 = (11 - 7.5) / 4.53 ≈ 0.77
z3 = (10 - 7.5) / 4.53 ≈ 0.55
z4 = (6 - 7.5) / 4.53 ≈ -0.33
z5 = (6 - 7.5) / 4.53 ≈ -0.33
z6 = (15 - 7.5) / 4.53 ≈ 1.65
z7 = (2 - 7.5) / 4.53 ≈ -1.22
z8 = (15 - 7.5) / 4.53 ≈ 1.65
z9 = (3 - 7.5) / 4.53 ≈ -0.99
z10 = (2 - 7.5) / 4.53 ≈ -1.22

We can consider any z-score that is greater than 3 or less than -3 as an outlier.

There are no z-scores in this data set that are greater than 3 or less than -3. Therefore, there are no outliers in the data.