Question

Often frequency data are tabulated according to two criteria, with a view toward testing whether the criteria are associated. Consider the following analysis of the 157 machine breakdowns during a given quarter. Number of Breakdowns We are interested in whether the same percentage of breakdown occurs on each machine during each shift or whether there is some difference due perhaps to untrained operators or other factors peculiar to a given shift.

Short answer

To determine whether the same percentage of breakdowns occurs on Machines A, B, and C during each shift, we organized the data in a contingency table and calculated the expected frequency for each cell. Next, we calculated the chi-square statistic and determined the degrees of freedom. Finally, we compared the chi-square statistic with the critical value from a chi-square distribution table to test the hypothesis of association between machines and shifts. If the calculated chi-square statistic is greater than the critical value, there is a significant association between machines and shifts, indicating a difference in breakdown percentages. If not, there is no association, suggesting the same percentage of breakdowns occurs on each machine during each shift.

Step-by-step solution

Organize the data in a contingency table

Let's organize the given data in a contingency table. We will have three machines as rows (Machine A, Machine B, and Machine C) and three shifts as columns (Shift 1, Shift 2, and Shift 3). The cells represent the number of breakdowns for each machine on different shifts.

Calculate the row and column totals

To determine whether the same percentage of breakdowns occur in each shift, we need to calculate the row totals (total breakdowns for each machine) and column totals (total breakdowns for each shift).

Calculate the expected frequency for each cell

The expected frequency for each cell is calculated by multiplying the row total for that machine by the column total for that shift, then dividing by the total number of breakdowns. Expected frequency = (Row total × Column total) / Total number of breakdowns

Calculate the chi-square statistic using the observed and expected frequencies

Using the observed and expected frequencies, we can calculate the chi-square statistic: \( \chi^2 = \Sigma \frac{(Observed - Expected)^2}{Expected} \) Where the sum is taken over all the cells of the contingency table.

Determine the degrees of freedom

To determine the degrees of freedom for this problem, we can use the formula: Degrees of freedom = (Number of rows - 1) × (Number of columns - 1) In our case, there are 3 rows (machines) and 3 columns (shifts), so the degrees of freedom will be: Degrees of freedom = (3 - 1) × (3 - 1) = 2 × 2 = 4

Test the hypothesis of association using the chi-square statistic and the degrees of freedom

Now that we have calculated the chi-square statistic and the degrees of freedom, we can use these values to test the hypothesis of association between machines and shifts. We will compare the calculated chi-square statistic with the critical value from a chi-square distribution table for significance level (typically α = 0.05) and the degrees of freedom. If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant association between machines and shifts. If the calculated chi-square statistic is not greater than the critical value, we fail to reject the null hypothesis and conclude that there is no association between machines and shifts. By following these steps, you will be able to determine whether the same percentage of breakdowns occurs on Machines A, B, and C during each shift or if there is a difference due to factors such as untrained operators.

Key concepts

Contingency Table

The foundation of the chi-square test lies in the construction of a contingency table, which organizes categorical data for two variables into a matrix format. This table is instrumental when you're interested in exploring the relationship or association between these two variables.

A typical contingency table consists of rows and columns with cells representing the frequency count of data points that fall into the corresponding category of each variable. In the example provided, the contingency table would organize 'Machine' (A, B, or C) across the rows and 'Shift' (1, 2, or 3) across the columns. The intersection cells contain the number of breakdowns for each machine-shift pairing.

Creating a clear and informative contingency table is the first step because it provides a visual summary of the data and is the basis from which expected frequencies and the chi-square statistic are calculated.

Expected Frequency

A key component of the chi-square test is calculating the expected frequencies. This refers to the number of occurrences we would expect for each cell in the contingency table if no association exists between the row and column variables—in other words, if they were independent.

To find the expected frequency, the formula is as follows: \( \text{Expected frequency} = \frac{(\text{Row total} \times \text{Column total})}{\text{Total number of observations}} \)

These expected values are crucial because they serve as a benchmark against which the observed frequencies are compared. The chi-square statistic is essentially a sum of the squared differences between observed and expected frequencies, normalized by the expected frequencies.

Degrees of Freedom

When performing a chi-square test, calculating the degrees of freedom (df) is an essential step. Degrees of freedom refers to the number of independent values or quantities which can vary in an analysis without breaking any constraints. In the context of a contingency table, it's calculated based on the number of categories in each variable.

The formula for degrees of freedom in a contingency table is: \( df = (\text{Number of rows} - 1) \times (\text{Number of columns} - 1) \)

In the provided example, with 3 machines and 3 shifts, there are \( (3-1) \times (3-1) = 4 \) df. This value is important because it determines the shape of the chi-square distribution used to evaluate the significance of the test statistic.

Hypothesis Testing

Hypothesis testing is the process of making inferences about a population based on sample data. In the context of the chi-square test, it involves two hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)).

The null hypothesis asserts there is no association between the variables—in this case, machine and shift types—meaning all machines would have a similar breakdown frequency during shifts. The alternative hypothesis claims there is an association, implying that the machine breakdowns are influenced by the shifts.

To test these hypotheses, one compares the calculated chi-square statistic with a critical value from the chi-square distribution table. If the statistic exceeds the critical value at the chosen significance level (\(\alpha\)), often 0.05 or 5%, the null hypothesis is rejected in favor of the alternative. Therefore, the conclusion would be that a significant association exists between the variables.

Chi-Square Distribution

The chi-square distribution is an asymmetrical distribution that is used extensively in hypothesis testing, specifically in tests of independence like the chi-square test for contingency tables. The shape of a chi-square distribution varies depending on the degrees of freedom.

As df increase, the distribution becomes more symmetrical and resembles a normal distribution. In hypothesis testing, after calculating the chi-square statistic, one refers to the chi-square distribution corresponding to the calculated df. This distribution provides critical values that set thresholds for determining whether the observed frequencies differ significantly from the expected frequencies under the null hypothesis.

By comparing the calculated chi-square value with the critical value from this distribution, researchers can assess the statistical significance of their findings and draw conclusions about the association between the categorical variables.