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Student[Statistics][ShapiroWilkWTest] Overview

overview of the Shapiro Wilks W-Test

Description

 • Shapiro and Wilk's W-test is a test for normality. The Shapiro Wilk test tests the null hypothesis that a sample follows a normal distribution.
 • The formula of the test statistic is:

$W=\frac{{\left(\sum _{i=1}^{n}{a}_{i}X\left(i\right)\right)}^{2}}{\left(n-1\right)\mathrm{Variance}\left(X\right)}$

 where $X$ is the studied sample, $X\left(i\right)$ is the $i$th smallest data in X, ${X}_{i}$ is the $i$th data in X, ${a}_{i}$ are the coefficients to estimate straightness of the quantile-quantile plot.
 The definitions of these coefficients are beyond the scope of this guide.
 • The null hypothesis that the sample follows a normal distribution is rejected if W is too small.

Examples

Pete wants to use a one sample t-test to test the mean of the average lifetime of light bulbs of a particular type, but he does not know if the observations are normally distributed. To test this, he applies Shapiro and Wilk's W-test to the sample of data.

His observed data:

 bulb1 bulb2 bulb3 bulb4 bulb5 bulb6 bulb7 bulb8 bulb9 bulb10 lifetime(hrs) 355.0 359.5 379.3 366.5 325.1 334.4 308.4 355.6 381.2 316.9 bulb11 bulb12 bulb13 bulb14 bulb15 bulb16 bulb17 bulb18 bulb19 bulb20 lifetime(hrs) 379.0 338.7 380.3 366.4 368.1 333.3 390.7 337.4 373.3 370.0

Determine the null hypothesis:

 Null hypothesis: The data is normally distributed

Collect the data:

 > $X≔\left[355.0,359.5,379.3,366.5,325.1,334.4,308.4,355.6,381.2,316.9,379.0,338.7,380.3,366.4,368.1,333.3,390.7,337.4,373.3,370.0\right]:$

Run the Shapiro Wilk w-Test:

 > $\mathrm{Student}:-\mathrm{Statistics}:-\mathrm{ShapiroWilkWTest}\left(X\right):$
 Shapiro and Wilk's W-Test for Normality --------------------------------------- Null Hypothesis: Sample drawn from a population that follows a normal distribution Alt. Hypothesis: Sample drawn from population that does not follow a normal distribution   Sample Size:             20 Computed Statistic:      .935508635130523 Computed p-value:        .207505438819378   Result: [Accepted] This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

The Shapiro and Wilk's W-test returns a p-value = 0.207505. From this p-value, Pete concludes that the data can indeed be assumed to be normal and proceed with one sample t-test.