Chi-Square Goodness-of-Fit Test

Chi-Square Goodness-of-Fit Test: Using a Chi-Square Goodness of Fit Test with a significance

level of 0.05, test the hypothesis that Set 1 is sampled from a Normal Distribution with a population

mean equal to the sample mean and a population standard deviation equal to the sample standard

deviation. Similarly, test the hypothesis with a significance level of 0.05 that Set 2 is sampled from

an Exponential Distribution with a population mean equal to the sample mean. For each test, start

with the data classes from your histogram and merge them to ensure each class has a sufficient

number of observations. Then, for each data class, calculate the following:

• Numbers of observations in the data.

• Class probability.

• Class expected value.

• Chi-square component values.

Finally, for each test, calculate the chi-square value, describe the degrees of freedom, and explain

your conclusion.

Report: The homework report is to be typewritten in clear English with complete sentences. Be sure

to define all notations and include descriptions of all tables and figures in the text. Your report

should include a cover page and the following additional section: 

Goodness-of-Fit Tests: Describe the chi-square tests with tables for the calculated values and

clearly stated conclusions. Show the Excel formulas for your table calculations in an

Appendix.

IE 5317: Goodness-of-Fit Calculations

Construct an expanded frequency table by adding the columns for
(i) class probability,
(ii) class expected value, and
(iii) class chi-square component values.

Below the table, show the probability calculations for the first two or three classes.

Perform a goodness of fit test to determine if your data fits the specified distribution.

EXAMPLE

Class Observed
Frequency (oi)

Class Probability Expected
Frequency (ei)

χ2 Class
Component

X ≤ 2 Count
observations
based on your
collected data.

Calculate using
the assumed
probability
distribution.

For each class,
take its
probability and
multiply by n.

2 < X ≤ 7 7 < X ≤ 12 X > 12
Total n 1.0 n χ2 statistic

The class probabilities are:
P[X ≤ 2]
P[2 < X ≤ 7] = P[X ≤ 7] − P[X ≤ 2] P[7 < X ≤ 12] = P[X ≤ 12] − P[X ≤ 7] P[X > 12] = 1 − P[X ≤ 12]

For the normal distribution, you need 2 parameters: use the sample mean for μ and the sample
standard deviation for σ.
For the exponential, you need β: use the sample mean for β.

In Excel, to calculate c.d.f. values P[X ≤ x]: use NORMDIST(x, μ, σ, 1) for the normal
distribution and GAMMADIST(x, 1, β, 1) for the exponential distribution.

Finally, if any of the Expected Frequencies (ei) are less than 5, then you should combine
classes accordingly.

2( )i i
i

o e
e
−

Normal
Class Frequency Class Probability Expected Value Chi-square

x ≤ 15 17 0.101049773 10.10497729 4.704744689
15 < x ≤ 20 15 0.172503584 17.25035841 0.293565666 20 < x ≤ 25 2 0.254902993 25.49029933 21.64722177 25 < x ≤ 30 44 0.243371959 24.33719594 15.88621238 30 < x ≤ 35 18 0.150130721 15.01307207 0.594264681 x > 35 4 0.07804097 7.804096969 1.854302145

Sum 100 1 100 44.98031133 Chi-square test statistic value

Class Probability Calculations:
x ≤ 15 NORMDIST(15,24.47,7.424,1)
15 < x ≤ 20 NORMDIST(20,24.47,7.424,1) - NORMDIST(15,24.47,7.424,1) 20 < x ≤ 25 NORMDIST(25,24.47,7.424,1) - NORMDIST(20,24.47,7.424,1) 25 < x ≤ 30 NORMDIST(30,24.47,7.424,1) - NORMDIST(25,24.47,7.424,1) 30 < x ≤ 35 NORMDIST(35,24.47,7.424,1) - NORMDIST(30,24.47,7.424,1) x > 35 1 – NORMDIST(35,24.47,7.424,1)

Exponential
Class Frequency Class Probability Expected Value Chi-square

x ≤ 150 30 0.432229355 43.2229355 4.045213988
150 < x ≤ 300 40 0.24540714 24.54071397 9.738491102 300 < x ≤ 450 17 0.13933497 13.933497 0.674880158 450 < x ≤ 600 4 0.079110306 7.911030577 1.933523076 x > 600 9 0.10391823 10.39182296 0.186413024

Sum 100 1 100 16.57852135 Chi-square test statistic value

Class Probability Calculations:
x ≤ 150 GAMMADIST(150,1,265,1)
150 < x ≤ 300 GAMMADIST(300,1,265,1) - GAMMADIST(150,1,265,1) 300 < x ≤ 450 GAMMADIST(450,1,265,1) - GAMMADIST(300,1,265,1) 450 < x ≤ 600 GAMMADIST(600,1,265,1) - GAMMADIST(450,1,265,1) x > 600 1 – GAMMADIST(600,1,265,1)

• 5317-goodness-of-fit-example

3301-Project2-calculations
Sheet1