Problem 5 - An Introduction to Psychological Statistics

Question

Compute the expected frequencies for the following contingency table: $$ \begin{array}{lcc} & \text { Category A } & \text { Category B } \\ \hline \text { Category C } & 22 & 38 \\ \text { Category D } & 16 & 14 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The expected frequencies are: Category C & A = 25.33, Category C & B = 34.67, Category D & A = 12.67, Category D & B = 17.33.

Step 1: Calculate Row Totals

Start by calculating the total for each row in the table. For Category C, the row total is: \( 22 + 38 = 60 \).For Category D, the row total is: \( 16 + 14 = 30 \).

Step 2: Calculate Column Totals

Next, find the total for each column in the table. For Category A, the column total is: \( 22 + 16 = 38 \).For Category B, the column total is: \( 38 + 14 = 52 \).

Step 3: Compute Total Frequency

Now, calculate the overall total frequency by adding all the entries in the table: \( 22 + 38 + 16 + 14 = 90 \). The grand total is 90.

Step 4: Compute Expected Frequency for Category C & A

The expected frequency for each cell in a contingency table can be calculated using the formula:\[E_{ij} = \frac{( ext{Row Total})_i \times ( ext{Column Total})_j}{ ext{Grand Total}}\]For Category C & A: \[E_{11} = \frac{60 \times 38}{90} = 25.33\]

Step 5: Compute Expected Frequency for Category C & B

For Category C & B:\[E_{12} = \frac{60 \times 52}{90} = 34.67\]

Step 6: Compute Expected Frequency for Category D & A

For Category D & A:\[E_{21} = \frac{30 \times 38}{90} = 12.67\]

Step 7: Compute Expected Frequency for Category D & B

For Category D & B:\[E_{22} = \frac{30 \times 52}{90} = 17.33\]

Step 8: Compile Expected Frequencies

Compile the calculated expected frequencies into a table:\[\begin{array}{lcc}& \text{Category A} & \text{Category B} \\hline \text{Category C} & 25.33 & 34.67 \\text{Category D} & 12.67 & 17.33 \\hline\end{array}\]

Key Concepts

Expected Frequencies

Expected frequencies are crucial in understanding how different categories in a contingency table ideally distribute based on statistical assumptions. These expected values help in comparing what we see in our data to what we would expect if there were no relationship between the variables.

To compute the expected frequencies for each cell in a contingency table, you need to use the formula:

• Identify the total of each row and each column.
• Calculate the grand total, which is the sum of all the entries in the table.
• Use the formula: \(E_{ij} = \frac{(\text{Row Total})_i \times (\text{Column Total})_j}{\text{Grand Total}}\) to find the expected frequency for cell \(ij\).

This formula helps allocate the expected cell counts by considering how much of the grand total comes from each row and column. The resulting expected frequencies provide a theoretical distribution if categories were independent of each other.

Chi-Square Test

The Chi-Square Test is a statistical method used to determine if there is a significant association between categorical variables in a contingency table. By comparing the observed frequencies in the data with the expected frequencies calculated from the distribution assumptions, it assesses whether any observed differences could be due to chance.

The steps involved in conducting a Chi-Square Test include:

• Setting up a null hypothesis stating that the variables are independent.
• Calculating the expected frequencies for comparisons.
• Computing the Chi-Square statistic using the formula: \[\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}\]where \(O_{ij}\) represents the observed frequency in cell \(ij\) and \(E_{ij}\) is the expected frequency.
• Comparing the calculated Chi-Square statistic to a critical value from a Chi-Square distribution table to determine significance.

This test is advantageous for evaluating whether the observed data significantly deviates from what was expected under independence.

Categorical Data Analysis

Categorical data analysis concerns the investigation and interpretation of data that can be classified into categories, often displayed in a contingency table format. It is an essential branch of statistics, especially useful in hypothesis testing and understanding relationships between variables.

Key procedures in handling categorical data include:

• Creating contingency tables to organize and summarize the data.
• Calculating expected frequencies and using statistical tests like the Chi-Square Test to examine relationships.
• Interpreting results to understand patterns and implications of data.

This type of analysis enables researchers to decide whether associations in data are statistically significant or simply the result of random variability. It turns raw data into actionable insights by applying statistical techniques, ensuring that interpretations of relationships between categories are accurate and meaningful.