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Let the result of a random experiment be classified by two attributes. Let these two attributes be eye color and hair color. One attribute of the outcome, eye color, can be divided into certain mutually exclusive and exhaustive events. Let these events be: \(\mathrm{A}_{1}:\) Blue eyes \(\mathrm{A}_{2}:\) Brown eyes \(\mathrm{A}_{3}:\) Grey eyes \(\mathrm{A}_{4}:\) Black eyes \(\quad \mathrm{A}_{5}:\) Green eyes The other attribute of the outcome can also be divided into certain mutually exclusive and exhaustive events. Let these events be: \(\mathrm{B}_{1}:\) Blond hair \(\mathrm{B}_{2}:\) Brown hair \(\quad \mathrm{B}_{3}:\) Black hair \(\mathrm{B}_{4}:\) Red hair. The experiment is performed by observing \(\mathrm{n}=500\) people and each of them are categorized according to eye color and hair color. Let \(\mathrm{A}_{\mathrm{i}} \cap \mathrm{B}_{\mathrm{j}}\) be the event that a person with eye color \(\mathrm{A}_{\mathrm{i}}\) and hair color \(\mathrm{B}_{\mathrm{i}}\) is observed, \(\mathrm{i}=1,2,3,4,5\) and \(j=1,2,3,4\). Let \(X_{i j}\) be the observed frequency of event \(\mathrm{A}_{\mathrm{i}} \cap \mathrm{B}_{\mathrm{j}}\). Test the hypothesis that \(\mathrm{A}_{\mathrm{i}}\) and \(\mathrm{B}_{\mathrm{j}}\) are independent attributes.

Step by step solution

01

Create a Contingency Table

First, we must create a contingency table to summarize the observed frequencies of the combined events \(A_i \cap B_j\). This table will have five rows (one for each eye color event) and four columns (one for each hair color event).

02

Calculate Row and Column Totals

Next, calculate the row and column totals for the contingency table. This will involve summing the observed frequencies for each eye color event in each row and for each hair color event in each column.

03

Calculate Expected Frequencies

Now, calculate the expected frequency for each cell in the contingency table. The expected frequency for a cell \((i, j)\) is given by the formula: \( E_{ij} = \frac{(row \thinspace total_{i}) \times (col \thinspace total_{j})}{n} \), where \(n\) is the total number of observations (500 in our case).

04

Compute the Chi-Squared Test Statistic

Next, compute the chi-squared test statistic, denoted by \( \chi^2 \), using the following formula: \( \chi^2 = \sum_{i=1}^5 \sum_{j=1}^4 \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \), where \( O_{ij} \) are the observed frequencies and \( E_{ij} \) are the expected frequencies.

05

Determine the Degrees of Freedom

In order to compare our test statistic against a critical value, we need to determine the degrees of freedom. This is given by the formula: \( df = (number \thinspace of \thinspace rows - 1) \times (number \thinspace of \thinspace columns - 1) \), In our case, this gives us \(df = (5-1) \times (4-1) = 12\).

06

Find the Critical Value

Using a chi-squared distribution table, find the critical value for our calculated degrees of freedom (12) and a chosen significance level (e.g., 0.05).

07

Comparing the Test Statistic to the Critical Value

Compare the chi-squared test statistic to the critical value found in Step 6. If the test statistic is greater than the critical value, reject the null hypothesis that the eye color and hair color are independent attributes. Otherwise, we cannot reject the null hypothesis. In conclusion, by following these steps and applying the chi-squared test of independence with the given data, we can determine whether eye color and hair color are independent attributes in the observed population.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Contingency Table

A contingency table is a simple and organized way to present data that shows the frequency distribution of variables. In this case, the table categorizes individuals based on two attributes: eye color and hair color.

The table is structured with rows and columns. Each row represents a distinct category of eye color, and each column represents a category of hair color. This organization helps to visualize how the different categories interact with each other.

For the given problem, the contingency table consists of five rows and four columns. That's because there are five types of eye colors and four types of hair colors. By summarizing the data in this way, it becomes easier to see patterns or relationships between the attributes.

Most importantly, the contingency table is crucial for the next steps in the chi-squared test, as it helps compute both observed and expected frequencies.

Expected Frequency

Expected frequency helps us understand what we would anticipate if there were no association between categories. To compute the expected frequency for each cell in the contingency table, we use the formula: \[ E_{ij} = \frac{(\text{row total}_i) \times (\text{column total}_j)}{n} \] where \(E_{ij}\) represents the expected frequency of the cell at row \(i\) and column \(j\), and \(n\) is the total number of observations (500 in this case).

Calculating this value involves:

• Multiplying the total number of occurrences in a specific row by the total number of occurrences in a specific column.
• Dividing that product by the overall number of cases (total sample size).

These expected frequencies assume that the two attributes are independent. By comparing them to the observed frequencies, the chi-squared test determines whether the discrepancies are due to chance or indicate a possible relationship between the variables.

Degrees of Freedom

The degrees of freedom in a statistical test, like the chi-squared test, help indicate how many independent values or quantities can vary in the analysis. For a contingency table, the degrees of freedom are calculated using the formula:\[ df = (\text{number of rows} - 1) \times (\text{number of columns} - 1) \]In our example, we have a 5x4 table, leading to a calculation of:\[ df = (5-1) \times (4-1) = 12 \]

These degrees of freedom allow us to find the critical value in the chi-squared distribution table. This critical value serves as a threshold to decide whether to accept or reject the null hypothesis. More degrees of freedom typically mean more complex models, as they represent more categories and potential variability in the dataset.