R Language for The Project Management Course — AUG University “part 7”

Welcome back! This is the 7th chapter of the series explaining the project management course for AUG students — The computer system engineering department.

Anderson-Darling Test(A²):

AD-Test is used to determine used to test if a sample of data came from a population with a specific distribution. Usually, used for normal distribution Therefore, we can use the same rules we use in the normal distribution.

The hypotheses for the AD-test are:

• H0: The data comes from a specified distribution.
• H1: The data does not come from a specified distribution.

Perform AD by Hand:

1. Using this formula, calculate the AD Statistic:

Note: n is the sample size, F(x) is the CDF function, i is the ith sample that you calculated by sorting the data in ascending order.

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2. Find the statistic’s p-value.

The formula for the p-value depends on the value for the AD statistic from Step 1.

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Small p-values (less than your chosen alpha level) mean that you can reject the null hypothesis. In other words, the data does not come from the named distribution.

AD-Test Example in Excel:

Step1: Enter your data like the following (page 25 in the slides):

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Step 2: calculate 2i -1

A. Select the cell you want to calculate its value

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B. Enter “=” and select the A2 cell which contains “1"

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C. Add to the cell you selected “*2 -1”

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D. Tap “Enter” so the result will show on.

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E. select all the cells then tap on Crtl +Enter so all the cells are calculated automatically.

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Step3: Calculate ln(f(xi)), — x is A-Z order —

first, calculate Xi (A-Z) and Xi(Z-A)

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then, calculate mean, standard deviation, and x

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After that, calculate f(x)using Norm.Dist(x, mean, standard deviation, cumulative) for each cell. X is from A-Z

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Finally, calculate ln(f(X))

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Step3: Calculate ln(1-f(x)): x is Z-A order

first, calculate f(x), x from Z-A then calculate ln(1-f(x))

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Step4: Calculate (2i-1) *( ln(f(x))+ ln(1-F(Xn-i+1)))

Calculate (2i-1) *( ln(f(x))+ ln(1-F(Xn-i+1)))

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Step5: Caluclate S value

Using this formula:

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first, calculate ((2i−1)/ n) * [lnF(Yi)+ln(1−F(YN+1−i))]

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Then, we will do the sum for all cells

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Step6: Calculate A² value

Using this formala :A² = |-n -S|.

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Step6: Calculate A² critical value

• From this table, choose the nearest n value to your n data which is 9. This means we choose n =10
• After that, find the cross cell between %5 and n =10
• A² critical value is 0.683

Note: Significance level α usually = %5, this means if we repeat the experiment 100 times, we must have the same results in 95 times, and we are allowed to have a different result just in 5 times. The 5 different results = α

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Step7: Compare A² with the A² critical value

We will compare A² with the A² critical value, if

• If A² < A² critical value then, you can consider your data distributed as a normal distribution.
• If A² > A² critical value then, your data is not distributed as a normal distribution.

A² = 0.219 and A² critical value = 0.683. Since 0.219 < 0.683 then, your data is distributed as a normal distribution.

or

you can calculate the p-value, then if p-value > α failed to reject.

You can access the data I worked on by clicking on this sheet.

AD-Test in R:

Usage

ad.test(x)

you can see the R Documentation for more details.

See you in the next part.

Resources

Anderson-Darling Test & Statistic: Definition, Examples

1.3.5.14. Anderson-Darling Test

ad.test function - RDocumentation