Testing for Normality

The Shapiro-Wilk Test allows us to test the claim that a sample comes from a normally distributed population, an assumption for many hypothesis tests.  This test always has the same hypotheses:

H₀: The sample comes from a normally distributed population
H₁: The sample does not come from a normally distributed population

In order to conduct the Shapiro-Wilk test we will apply the command shapiro.test to any data set.

Example: The table below provides the data on the number of cats and dogs adopted from Paws Chicago by year.  Test the claim that this sample data comes from a population that is normally distributed?  Use α = 0.05.

2011	2010	2009	2008	2007	2006
4268	4042	3467	3015	1666	946

Source: www.pawschicago.org

We need to begin by creating a data set.

> paws = c(4268, 4042, 3467, 3015, 1666, 946)

We can now conduct the Shapiro-Wilk Test to test the claim.

> shapiro.test(paws)

        Shapiro-Wilk normality test

data:  paws 
W = 0.9131, p-value = 0.4568

Since our p-value = 0.4568 > 0.05 = α we fail to reject H₀.  The data supports the claim that this sample comes from a normally distributed population.

Example: The tweets of 11 casinos were tracked during the week of June 1 - June 7, 2010.  The table below provides the number of positive interactions each casino had with customers during that week.  Test the claim that this sample data comes from a normally distributed population. Use α = 0.02.

Casino	Number of Positive Interactions
Aria	1
Caesars Palace	12
Casino Royale	0
Excalibur	21
Hard Rock Hotel	2
Las Vegas Hilton	6
Mirage	59
Planet Hollywood	21
Station Casinos	7
Venetian	4
Wynn Las Vegas	42

Source: gaming.unlv.edu

Again, we start by creating a data set.

> pos.int = c(1, 12, 0, 21, 2, 6, 59, 21, 7, 4, 42)

We can now conduct the Shapiro-Wilk Test to test the claim.

> shapiro.test(pos.int)

        Shapiro-Wilk normality test

data:  pos.int 
W = 0.805, p-value = 0.01096

Since our p-value = 0.01096 ≤ 0.02 = α we reject H₀.  The data does not support the claim that this sample comes from a normally distributed population.