Chi-Square Tests, Goodness of Fit Test Problem -> https://youtu.be/5L4balz_bp8

Chi-square tests are a family of statistical tests that assess the relationship between categorical variables. These tests compare the observed frequencies in different categories with the frequencies that would be expected if there were no association or difference between the variables. The chi-square test calculates a test statistic (chi-square statistic) based on the differences between observed and expected frequencies, and determines whether the differences are statistically significant.

Now let's consider a numerical problem to demonstrate a goodness-of-fit test, which is a specific type of chi-square test.

Suppose you are interested in examining whether the distribution of favorite ice cream flavors in a population matches a specific distribution stated by an ice cream company. The company claims that the population distribution should be 30% chocolate, 40% vanilla, and 30% strawberry.

To test this claim, you collect data from a random sample of 200 individuals and ask them to choose their favorite ice cream flavor. The observed frequencies for each flavor in your sample are as follows: 65 individuals prefer chocolate, 80 prefer vanilla, and 55 prefer strawberry.

To conduct a goodness-of-fit test, you need to determine whether the observed frequencies significantly deviate from the expected frequencies based on the company's distribution.

Step 1: Define hypotheses:

• Null hypothesis (H₀): The distribution of ice cream flavors in the population follows the company's stated distribution.
• Alternative hypothesis (H₁): The distribution of ice cream flavors in the population does not follow the company's stated distribution.

Step 2: Set the significance level (α): Choose a significance level, such as α = 0.05, to determine the level of evidence required to reject the null hypothesis.

Step 3: Calculate expected frequencies: Based on the company's stated distribution, calculate the expected frequencies for each flavor. In this case, the expected frequencies are:

• Chocolate: 0.30 * 200 = 60
• Vanilla: 0.40 * 200 = 80
• Strawberry: 0.30 * 200 = 60

Step 4: Calculate the chi-square statistic: The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies. The formula is: χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ], where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each category.

Using the observed and expected frequencies, the calculation is as follows: χ² = [(65-60)²/60] + [(80-80)²/80] + [(55-60)²/60] = 50/60= 0.83333

Step 5: Determine the critical value and p-value: The chi-square statistic follows a chi-square distribution. With two degrees of freedom (three categories - 1), you can consult a chi-square distribution table or use statistical software to find the critical value associated with α = 0.05. Additionally, the software can provide the p-value, which represents the probability of observing a chi-square statistic as extreme as the calculated value under the null hypothesis.

Step 6: Make a decision: If the calculated chi-square statistic exceeds the critical value or the p-value is less than the chosen significance level (α), you reject the null hypothesis. This indicates that there is evidence to suggest the distribution of ice cream flavors in the population does not match the company's stated distribution.

In summary, a goodness-of-fit test using a chi-square test allows you to determine whether observed frequencies significantly differ from expected frequencies. By comparing the calculated chi-square statistic to critical values or using the p-value, you can draw conclusions about the distribution and assess the claim made by the company.

Remember that this example is simplified and serves to illustrate the process. In practice, data analysis often involves more complex scenarios, sample sizes, and categorical variables.

A chi-square test is a statistical test that is used to compare observed and expected results. The goal of this test is to identify whether a disparity between actual and predicted data is due to chance or to a link between the variables under consideration. There are two types of chi-square tests: the chi-square goodness of fit test and the chi-square test of independence.

The chi-square goodness of fit test is used to test whether the frequency distribution of a categorical variable is different from your expectations. For example, you can use this test to check if a six-sided die is fair by rolling it many times and comparing the observed frequencies of each face with the expected frequencies (which should be equal for a fair die).

To perform a chi-square goodness of fit test, you need to calculate the chi-square statistic, which is given by the formula:

chi-square = sum of (observed - expected)^2 / expected

where observed is the frequency of each category in your data, and expected is the frequency of each category according to your hypothesis. The higher the chi-square value, the more the observed frequencies deviate from the expected frequencies.

To determine whether the difference between observed and expected frequencies is statistically significant, you need to compare the chi-square value with a critical value from a chi-square distribution table. The critical value depends on the level of significance (usually 0.05) and the degrees of freedom, which is equal to the number of categories minus one.

If the chi-square value is greater than or equal to the critical value, you reject the null hypothesis that there is no difference between observed and expected frequencies. If the chi-square value is less than the critical value, you fail to reject the null hypothesis.

For example, suppose you roll a die 60 times and get the following results:

Face | Frequency

1 | 6

2 | 9

3 | 8

4 | 12

5 | 10

6 | 15

You want to test whether the die is fair using a chi-square goodness of fit test at a 0.05 level of significance. The null hypothesis is that there is no difference between observed and expected frequencies, which means that each face should have an expected frequency of 60 / 6 = 10.

The chi-square value is calculated as follows:

chi-square = (6 - 10)^2 / 10 + (9 - 10)^2 / 10 + ... + (15 - 10)^2 / 10

= 1.6 + 0.1 + 0.4 + 0.4 + 0 + 2.5

= 5

The degrees of freedom are 6 - 1 = 5. From a chi-square distribution table, the critical value for alpha = 0.05 and df = 5 is 11.07.

Since the chi-square value (5) is less than the critical value (11.07), you fail to reject the null hypothesis. There is not enough evidence to conclude that the die is unfair.