What do you mean by expected frequencies in (a) chi-square test for testing independence of attributes, and (b) chi-square test for testing goodness-of-fit? Also explain the procedure you follow in calculating the expected values in each of the above situations.

 Q. What do you mean by expected frequencies in (a) chi-square test for testing independence of attributes, and (b) chi-square test for testing goodness-of-fit? Also explain the procedure you follow in calculating the expected values in each of the above situations.

The chi-square test is a fundamental statistical test used to examine whether observed data fits a certain distribution or if two categorical variables are independent. In both the chi-square test for independence and the chi-square test for goodness-of-fit, expected frequencies are a crucial element. Understanding how expected frequencies are calculated and how they play into the overall hypothesis testing process is vital. Below is a detailed explanation of what expected frequencies mean in both contexts, followed by the steps involved in their calculation.


Chi-Square Test for Testing Independence of Attributes:

In the chi-square test for independence, the goal is to determine whether there is a significant association between two categorical variables, typically arranged in a contingency table. The test is based on comparing the observed frequencies in each category with the frequencies we would expect to see if the variables were independent.

Expected Frequencies in Chi-Square Test for Independence:

The expected frequency in this context represents the frequency that would occur in each cell of the contingency table if the two variables were independent. These expected frequencies are based on the assumption that the distribution of one variable does not depend on the distribution of the other. In other words, the expected frequencies reflect what we would expect to observe if the null hypothesis (that the two variables are independent) were true.

Formula for Expected Frequencies:

For a contingency table with rr rows and cc columns, the expected frequency for a cell in the ii-th row and jj-th column is calculated using the formula:

Eij=(Row totali)×(Column totalj)Grand totalE_{ij} = \frac{( \text{Row total}_i ) \times ( \text{Column total}_j )}{\text{Grand total}}Eij=Grand total(Row totali)×(Column totalj)

Where:

  • EijE_{ij}Eij is the expected frequency for the cell in the ii-th row and jj-th column.
  • Row total Row totali\text{Row total}_i is the total number of observations in the ii-th row.
  • Column total Column totalj\text{Column total}_j is the total number of observations in the jj-th column.
  • Grand total is the total number of observations across all cells.
  • The expected frequency for each cell reflects how many observations would be expected in that cell if the two variables were independent.

    Steps to Calculate Expected Frequencies in Chi-Square Test for Independence:

    1.    Create a Contingency Table: Arrange the data in a contingency table, where each cell represents a combination of categories from the two variables being studied.

    2.    Calculate Row Totals, Column Totals, and Grand Total:

    o   Find the total number of observations in each row and each column.

    o   Compute the grand total, which is the total number of observations across the entire table.

    3.    Apply the Formula: Use the formula for expected frequencies to calculate the expected number of observations for each cell. Multiply the row total by the column total, then divide by the grand total for each cell.

    4.    Compare Observed and Expected Frequencies: Once you have the expected frequencies, you can calculate the chi-square statistic by comparing the observed and expected frequencies for each cell. This involves summing the squared differences between the observed and expected frequencies, divided by the expected frequencies:

    χ2=i=1rj=1c(OijEij)2Eij\chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}χ2=i=1rj=1cEij(OijEij)2

    Where OijO_{ij} is the observed frequency for the cell and EijE_{ij} is the expected frequency for the cell.

    5.    Statistical Inference: Using the chi-square statistic and the appropriate degrees of freedom, you can then determine whether to reject or fail to reject the null hypothesis. If the chi-square statistic is large, it suggests that the observed frequencies differ significantly from the expected frequencies, which would lead to the rejection of the null hypothesis and conclude that the variables are not independent.

    Chi-Square Test for Testing Goodness-of-Fit:

    The chi-square test for goodness-of-fit is used to determine whether an observed distribution of data matches a specific theoretical distribution. This test is typically applied to a single categorical variable, where the goal is to assess if the observed frequencies across categories match the expected frequencies based on a known distribution.

    Expected Frequencies in Chi-Square Test for Goodness-of-Fit:

    In the chi-square goodness-of-fit test, the expected frequency for each category represents how many observations we would expect to see in that category if the data were to follow the specified distribution. For example, if we were testing whether a die is fair, the expected frequency for each of the six faces of the die would be the same (i.e., 16\frac{1}{6} of the total number of rolls).

    Formula for Expected Frequencies:

    The expected frequency for each category in a goodness-of-fit test is calculated by multiplying the total number of observations by the proportion that each category is expected to have according to the theoretical distribution. If we are testing whether data follows a uniform distribution with kk categories, the expected frequency for each category is given by:

    Ei=nkE_i = \frac{n}{k}Ei=kn

    Where:

    • EiE_iEi is the expected frequency for the ii-th category.
    • nnn is the total number of observations.
    • kkk is the number of categories.

      If the distribution is not uniform, then each category may have a different expected frequency, and these frequencies are based on the theoretical probabilities for each category.

      Steps to Calculate Expected Frequencies in Chi-Square Test for Goodness-of-Fit:

      1.    Determine the Expected Distribution: Identify the theoretical distribution that the data is expected to follow. This could be a uniform distribution, a normal distribution, or any other known distribution.

      2.    Calculate Expected Frequencies: Multiply the total number of observations by the expected probability for each category. This gives you the expected frequency for each category.

      3.    Apply the Chi-Square Formula: Once the expected frequencies are determined, calculate the chi-square statistic by comparing the observed frequencies with the expected frequencies. The formula for the chi-square statistic in the goodness-of-fit test is:

      χ2=i=1k(OiEi)2Ei\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}χ2=i=1kEi(OiEi)2

      Where:

      • OiO_iOi is the observed frequency for the ii-th category.
      • EiE_iEi is the expected frequency for the ii-th category.
      • kkk is the number of categories.

        4.    Statistical Inference: Use the chi-square statistic and the degrees of freedom (which is k1k - 1 for a goodness-of-fit test) to determine whether the observed data significantly differs from the expected distribution. If the chi-square statistic is large, you may reject the null hypothesis, indicating that the observed data does not fit the expected distribution.

        Key Differences Between Expected Frequencies in Both Tests:

        1.    Context of Use:

        o   In the chi-square test for independence, expected frequencies are based on the assumption that the two categorical variables are independent of each other.

        o   In the chi-square test for goodness-of-fit, expected frequencies are based on a theoretical distribution or model, such as a uniform distribution or a distribution specified by the researcher.

        2.    Formula for Calculation:

        o   In the chi-square test for independence, expected frequencies are calculated using the formula Eij=(Row totali)×(Column totalj)Grand totalE_{ij} = \frac{( \text{Row total}_i ) \times ( \text{Column total}_j )}{\text{Grand total}}, which accounts for the marginal totals of the rows and columns.

        o   In the chi-square test for goodness-of-fit, expected frequencies are calculated by multiplying the total number of observations by the probability for each category according to the expected distribution.

        3.    Purpose:

        o   In the chi-square test for independence, the focus is on testing whether there is an association between two categorical variables.

        o   In the chi-square test for goodness-of-fit, the goal is to test whether the observed data conforms to a specified theoretical distribution.

        Conclusion:

        In both the chi-square test for testing independence of attributes and the chi-square test for testing goodness-of-fit, expected frequencies are central to the hypothesis testing process. In the test for independence, expected frequencies help assess whether the variables are independent, while in the goodness-of-fit test, expected frequencies determine how well the observed data matches a known distribution. The steps for calculating expected frequencies in each case involve using different formulas that reflect the nature of the hypothesis being tested. Understanding how to compute and interpret expected frequencies is essential for conducting chi-square tests effectively and drawing valid conclusions based on categorical data.

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