Are the Dice Fair?

Comparing Counts with the 𝛘² Goodness of Fit Test

a.k.a. The Badness-of-Grammar Test

Unit 5, 6, 7 Test

Handed back

Haven’t taken it? You have until 4/15 (tax day) to make it up

Find a VS

What common traits justify using a ꭓ² Goodness of Fit Test?

What type of variable(s) are you using?

𝛘2 Goodness of Fit Test Examples

To evaluate whether the distribution of stock price changes over time follows a particular expected distribution, such as the normal distribution.

To assess whether the observed genotype frequencies in a population match the expected frequencies under the Hardy-Weinberg equilibrium.

To check if the proportions of defective items produced by different machines in a factory are consistent with each other.

To evaluate whether the distribution of grades among students in different classes follows a particular expected distribution.

To analyze whether the proportions of people supporting different political parties in different regions are consistent with a theoretical distribution.

To determine if the ratings given by customers for different aspects of a service follow a particular expected distribution.

To examine whether the distribution of wins, losses, and ties for a sports team across multiple seasons matches a certain expected distribution.

To assess whether the distribution of different species in an ecosystem matches the expected distribution based on ecological models.

To investigate whether the distribution of blood types in a population matches the expected distribution based on genetic inheritance.

To determine if the distribution of preferences for different product features matches the expected distribution.

What common traits justify using a ꭓ² Goodness of Fit Test?

What type of variable(s) are you using?

𝛘2 Goodness of Fit Test Examples

To determine if the distribution of preferences for different product features matches the expected distribution.

To assess whether the observed genotype frequencies in a population match the expected frequencies under the Hardy-Weinberg equilibrium.

To check if the proportions of defective items produced by different machines in a factory are consistent with each other.

To evaluate whether the distribution of grades among students in different classes follows a particular expected distribution.

To analyze whether the proportions of people supporting different political parties in different regions are consistent with a theoretical distribution.

To determine if the ratings given by customers for different aspects of a service follow a particular expected distribution.

To examine whether the distribution of wins, losses, and ties for a sports team across multiple seasons matches a certain expected distribution.

To assess whether the distribution of different species in an ecosystem matches the expected distribution based on ecological models.

To investigate whether the distribution of blood types in a population matches the expected distribution based on genetic inheritance.

To evaluate whether the distribution of stock price changes over time follows a particular expected distribution, such as the normal distribution.

Important Ideas: 𝛘² Goodness of Fit Test

• There is no 𝛘² confidence interval. Only a hypothesis test… and it is always a one sided test.
• You have one categorical variable with multiple categories.
• You’re testing how well the observed distribution matches a expected distribution.
• The data consists of counts (not proportions or percentages).
• All expected counts are > 5

7

Fair Dice?

We use dice a lot in our math classes to learn various concepts… but are they fair?

Can we use the 𝛘² GOF test?

• The test is used when you have one categorical variable from a single population.
• This test is used to determine whether sample data are consistent with a hypothesized distribution.

8

DIY Loaded Dice

Hypotheses

If you were going to test this claim, what would your hypotheses be?

Write them in plain old English.

​

Ho:

​

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Ha:

Test Your 🎲

What’s the probability of that test statistic?

First we need to know the shape of the distribution.

Is it normal? Is it something else?

Play around with the 𝛘² distribution link on stapplet to find out.

df = number of categories - 1 (we have 6 categories)

12

Calculating the Test Statistic and p-Value

With your calculators

Enter your data into Lists first

• L1 → Observed Counts
• L2 → Expected Counts

Then,

• Stat → Tests → D: 𝛘² GOF-Test

Where

• df = number of categories - 1

(we have 6 categories)

14

Hypothesis Test

• Check
• Conditions
• Calculate
• Conclude

Big Ideas: 𝛘² Goodness of Fit Test

• State appropriate hypotheses and compute the expected counts and chi-square test statistic for a chi-squared Goodness of Fit Test.
• Calculate the degrees of freedom and p-value for a chi-squared Goodness of Fit Test.
• State and check the Random, 10%, and Large Counts conditions for performing a chi-squared Goodness of Fit Test.
• Perform a chi-squared Goodness of Fit Test.
• Conduct a follow-up analysis of which counts were the largest contributor to the chi-squared test statistic when the results of a chi-squared Test are statistically significant.