Download 12.1 Goodness-of

The Goodness-of-Fit Test
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Testing Claims about the Distribution of a Variable
Step 0: Verify Assumptions
The goodness-of-fit test has two requirements.
1. All expected counts are greater than or equal to 1 (Ei ≥ 1).
2. No more than 20% of the expected counts are less than 5.
Step 1: State the Hypothesis
A claim is made regarding a distribution based on a single population. This claim is used to
determine the following null and alternative hypotheses:
H0: The random variable follows the claimed distribution.
H1: The random variable does not follow the claimed distribution.
Step 2: Select a Level of Significance
The selection of the level of significance α is done based on the seriousness of making a Type I
error. (The typical value of α is 0.05.)
Step 3: Calculate the Test Statistic
The test statistic follows a χ2 distribution with k – 1 degrees of freedom.
1. Calculate the expected counts for each of the k categories. The expected counts are
Ei = µ i = npi for i = 1, 2, 3, …, k,
where n is the number of trials and pi is the probability of the ith category, assuming that
the null hypothesis is true.
2. The test statistics is given by:
χ 02 = ∑
(Oi − Ei )2
Ei
where Oi is the observed count for the ith category.
Step 4: Determine the Decision Criterion
The Classical Approach: Find the Critical Value
The level of significance is used to determine the critical value, represented by the χ2-value in
the figure below. All goodness-of-fit tests use a right-tailed test, so the critical value is χ α2
with k – 1 degrees of freedom.
χ α2
The Goodness-of-Fit Test
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The Modern Approach: Find the p-Value
Based on the test statistic χ 02 , determine the probability that the sum of the ratios of the
square of the difference between the observed and expected counts to the expected count is
more extreme than is found. This is represented by the shaded region under the chi square
distribution with k – 1 degrees of freedom in the figure below.
χ 02
Step 5: Make a Decision
Reject the null hypothesis if the test statistics lies in the critical region or the probability
associated with the test statistic is less than the level of significance.
Do not reject the null hypothesis if the test statistic does not lie in the critical region or the
probability associated with the test statistic is greater than or equal to the level of significance.
Step 6: State the Conclusion
State the conclusion of the hypothesis test based on the decision made and with respect to the
original claim.
Reject H0
Do Not
Reject H0
Original Claim is H0
There is sufficient evidence (at the α
level) to reject the claim that … .
There is not sufficient evidence (at the
α level) to reject the claim that … .
Original Claim is H1
There is sufficient evidence (at the α
level) to support the claim that … .
There is not sufficient evidence (at the
α level) to support the claim that … .