13. Chi-Square Tests & Goodness of Fit

Goodness of Fit Test - Excel

13. Chi-Square Tests & Goodness of Fit

Goodness of Fit Test - Excel: Videos & Practice Problems

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Performing a chi-square goodness of fit test in Excel involves setting hypotheses to compare observed and expected frequencies based on a claimed distribution. Calculate expected values using $n \times p$ , where $n$ is sample size and $p$ category proportion. Use Excel’s CHISQ.TEST function to find the p-value, which is compared to the significance level (alpha) to decide whether to reject the null hypothesis. This process aids in testing categorical data distributions efficiently, integrating key concepts like p-value, alpha level, and chi-square statistic.

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Goodness of FIt Test - Excel

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Goodness of FIt Test - Excel Video Summary

Performing a goodness of fit test in Excel streamlines the process of evaluating whether observed data matches a claimed distribution, especially when dealing with multiple categories and large sample sizes. This statistical test begins by formulating hypotheses: the null hypothesis assumes that the observed frequencies align with the claimed distribution, such as flavors being evenly distributed in a candy bag, while the alternative hypothesis states that the observed frequencies do not match the claim.

Key parameters include k, the number of categories, and n, the total sample size. For example, if there are eight flavors and 800 candies, then k = 8 and n = 800. When category proportions (p) are not provided, they can be calculated by dividing 1 by the number of categories, especially in cases of an even distribution, resulting in p = \frac{1}{k}. Using Excel, this calculation can be efficiently replicated across all categories.

Expected frequencies for each category are then computed by multiplying the total sample size by the category proportion, expressed as Expected = n \times p. Excel formulas allow for dynamic referencing, ensuring accuracy even if proportions vary across categories.

To determine the p-value, Excel’s CHISQ.TEST function is utilized, which compares the observed frequencies against the expected frequencies. The syntax is =CHISQ.TEST(actual_range, expected_range), where actual_range contains observed data and expected_range contains expected values. The resulting p-value quantifies the probability of observing the data if the null hypothesis were true.

Interpreting the p-value involves comparing it to the significance level (commonly α = 0.05). A p-value less than α indicates sufficient evidence to reject the null hypothesis, suggesting that the observed distribution significantly differs from the claimed distribution. For instance, a p-value of 0.00008 is much smaller than 0.05, leading to the conclusion that the flavors are not evenly distributed.

This approach highlights the importance of hypothesis testing in categorical data analysis and demonstrates how technology like Excel can simplify complex calculations, making statistical inference more accessible and efficient.

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Goodness of FIt Test - Excel Example 1

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Goodness of FIt Test - Excel Example 1 Video Summary

A goodness of fit test is used to determine whether observed frequencies in categorical data match an expected distribution. In this example, a 24-hour call center collected a random sample of 1,000 phone calls to assess if the proportion of calls during each four-hour window aligns with a claimed distribution. This test helps inform staffing decisions by verifying if call volumes are consistent with expectations.

The process begins by formulating hypotheses. The null hypothesis ($H_0$) states that the observed frequencies match the claimed distribution, while the alternative hypothesis ($H_a$) asserts that the observed frequencies do not match the claimed distribution. This sets the stage for a hypothesis test using a significance level of $\alpha = 0.1$.

Given the total sample size $n = 1000$ and the category proportions for each time window, the expected frequencies are calculated by multiplying the total sample size by each category proportion. For example, if a category proportion is $p_i$, the expected frequency $E_i$ is computed as:

\[E_i = n \times p_i\]

This calculation provides the expected counts under the null hypothesis for each category.

Next, the chi-square goodness of fit test statistic is used to compare observed and expected frequencies. The test statistic is calculated as:

\[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]

where $O_i$ represents the observed frequency for category $i$, and $E_i$ is the expected frequency. The p-value is then obtained from the chi-square distribution with degrees of freedom equal to the number of categories minus one.

In this case, the p-value was found to be 0.98, which is significantly greater than the significance level $\alpha = 0.1$. Since the p-value exceeds $\alpha$, there is insufficient evidence to reject the null hypothesis. This means the observed call frequencies do not significantly differ from the claimed distribution, supporting the assumption that call volumes are consistent across the specified time windows.

Understanding how to perform and interpret a chi-square goodness of fit test is essential for analyzing categorical data and making informed decisions based on observed versus expected distributions.

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To perform a chi-square goodness of fit test in Excel, start by setting your null hypothesis (H₀) that the observed frequencies match the claimed distribution, and the alternative hypothesis (H_a) that they do not. Next, calculate the expected frequencies by multiplying the total sample size $n$ by the category proportions $p$ . If the proportions are equal, $p = \frac{1}{k}$ , where $k$ is the number of categories. Then, use Excel's CHISQ.TEST function by selecting the observed frequencies range and the expected frequencies range as inputs. This function returns the p-value. Finally, compare the p-value to your significance level (alpha, usually 0.05). If the p-value is less than alpha, reject the null hypothesis, indicating the observed data does not fit the claimed distribution.

The expected value for each category in a goodness of fit test is calculated by multiplying the total sample size $n$ by the category proportion $p$ . The formula is: $E = n × p$ . In Excel, if n is in a cell (e.g., 800) and the category proportion p is in another cell, you can write =n_cell * p_cell to get the expected value. For example, if n is 800 and the category proportion is 0.125 (for equal distribution among 8 categories), the expected value is $800 × 0.125 = 100$ . This calculation is essential before running the chi-square test in Excel.

The p-value from Excel's CHISQ.TEST function indicates the probability of observing the data assuming the null hypothesis is true. To interpret it, compare the p-value to your significance level (alpha), commonly set at 0.05. If the p-value is less than alpha, it means the observed data significantly differs from the expected distribution, so you reject the null hypothesis. Conversely, if the p-value is greater than alpha, you fail to reject the null hypothesis, suggesting the observed data fits the claimed distribution. For example, a p-value of 0.00008 is much smaller than 0.05, so you would reject the null hypothesis and conclude the flavors are not evenly distributed.

When setting up hypotheses for a goodness of fit test in Excel, begin with the null hypothesis (H₀) stating that the observed frequencies match the claimed distribution. For example, if testing candy flavors, H₀ could be "flavors are evenly distributed." The alternative hypothesis (H_a) is that the observed frequencies do not match the claimed distribution, often phrased as "flavors are not evenly distributed." Writing clear hypotheses helps guide the test and interpretation. After setting these, calculate expected values and use Excel's CHISQ.TEST function to find the p-value for decision-making.

In Excel, to find the total sample size $n$ , you can use the SUM function to add all observed frequencies. For example, if observed frequencies are in cells A2 to A9, use =SUM(A2:A9). The number of categories $k$ is the count of distinct groups or flavors, which you can find by counting the rows or using the COUNTA function if categories are listed in a column. For instance, =COUNTA(B2:B9) counts the number of categories. These values are essential for calculating expected frequencies and performing the goodness of fit test.