Goodness of Fit Test: Videos & Practice Problems

A goodness of fit test evaluates whether observed frequencies align with expected frequencies based on a claimed distribution. The null hypothesis posits that these frequencies match, while the alternative suggests they do not. The test statistic, chi-squared (χ²), is calculated using the formula ${\chi }^2=\sum \frac{(}{O}/E$. A significant p-value indicates a poor fit, suggesting the observed data does not conform to the expected distribution.

Goodness of Fit Test

A goodness of fit test is a statistical method used to determine if the observed frequencies of a dataset align with the expected frequencies based on a specific distribution. This test is particularly useful when assessing whether a die, for example, is fair by comparing the actual outcomes of rolls to the theoretical outcomes expected from a uniform distribution.

In conducting a goodness of fit test, the first step involves formulating the hypotheses. The null hypothesis (H₀) posits that the observed frequencies match the expected frequencies, indicating that the distribution fits well. Conversely, the alternative hypothesis (H_a) suggests that at least one observed frequency differs from the expected frequency, implying a poor fit.

To illustrate, consider rolling a six-sided die 60 times. The expected frequency for each face of the die, assuming fairness, would be 10 (calculated as the total number of rolls divided by the number of categories: E = n/k, where n is the total rolls and k is the number of categories). The observed frequencies are the actual counts recorded from the rolls.

The test statistic for a goodness of fit test is calculated using the chi-squared statistic, represented as:

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

where O_i is the observed frequency and E_i is the expected frequency for each category. This formula quantifies the discrepancy between observed and expected frequencies across all categories.

After calculating the chi-squared statistic, the next step is to determine the p-value, which indicates the probability of observing the data if the null hypothesis is true. The degrees of freedom for this test are calculated as k - 1, where k is the number of categories. For our die example, with six categories, the degrees of freedom would be 5.

Using statistical tables or software, one can find the p-value corresponding to the calculated chi-squared statistic. If the p-value is less than the predetermined significance level (commonly set at α = 0.05), the null hypothesis is rejected, suggesting that the observed frequencies do not fit the expected distribution well. In our example, if the p-value is 0.0476, which is less than 0.05, we would reject the null hypothesis, concluding that the die is likely not fair.

When performing a goodness of fit test, it is essential to ensure that the sample is random, that all categories have observed frequencies, and that the expected frequencies are sufficiently large (typically at least 5) to validate the test's assumptions. Meeting these criteria ensures the reliability of the test results.

A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Write the null & alt. hypotheses to test the claim that the gym has equal numbers of members across all age groups.

• A. H₀: The # of members is the same for all age groupsH_a: The # of members is significantly different between the age groups
• B. H₀: The # of members is the same for all age groupsH_a: The # of members is significantly different for at least one of the age groups
• C. H₀: The # of members is significantly different for at least one of the age groupsH_a: The # of members is the same for all age groups
• D. H₀: The # of members is significantly different between the age groupsH_a: The # of members is the same for all age groups

Answer: (B)

A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Find the x² statistic to test the claim that the gym has equal numbers of members of all age ranges.

• A. 0.92
• B. 0.46
• C. 0.08
• D. 0.54

Answer: (A)

A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Using x² = 0.92 & α = 0.05, test the claim that the gym has equal numbers of members of all age ranges.

• A. Because P-value > α, we REJECT H₀. There is ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS NOT a good fit.
• B. Because P-value > α, we FAIL TO REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS a good fit.
• C. Because P-value < α, we REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS NOT a good fit.
• D. Because P-value < α, we FAIL TO REJECT H₀. There is NOT ENOUGH evidence that at the # of members is significantly different for at least one of the age groups at this gym. So the claimed dist. IS a good fit.

Answer: (B)

A gym owner wants to know if the gym has similar numbers of members across different age groups. The table shows the distribution of ages for members from a random survey. Does this data set fit the criteria for a G.O.F. test?

• A. Yes
• B. No
• C. More information is required.

Answer: (A)

Goodness of Fit Test Example 1

In this example, we explore the process of conducting a goodness of fit test to evaluate customer satisfaction survey responses across five categories: very poor, poor, neutral, good, and very good. The manager hypothesizes that the distribution of responses will not be uniform across these categories. To test this claim, a random sample of 100 survey responses is collected, and the observed frequencies for each category are recorded.

The first step in the analysis involves establishing the null and alternative hypotheses. The null hypothesis (H₀) posits that the frequencies for all rating categories are equal, indicating a uniform distribution of responses. Conversely, the alternative hypothesis (H_a) suggests that at least one category's frequency is significantly different from the others.

Next, we verify the conditions necessary for the goodness of fit test. We confirm that the sample is random, that there are observed frequencies for each category, and that the expected frequencies are sufficient (greater than or equal to five). The expected frequency for each category is calculated by dividing the total number of responses (n = 100) by the number of categories (k = 5), yielding an expected frequency of 20 for each category.

To compute the chi-squared test statistic (χ²), we use the formula:

χ² = ∑ (O_i - E_i)² / E_i

where O_i represents the observed frequencies and E_i represents the expected frequencies. By substituting the observed values into the formula, we calculate the chi-squared statistic, which results in a value of 10.3.

The degrees of freedom (df) for this test is determined by the formula df = k - 1, which in this case equals 4. Using the chi-squared statistic and the degrees of freedom, we find the p-value, which is calculated to be 0.0357.

To draw a conclusion, we compare the p-value to the significance level (α = 0.05). Since the p-value (0.0357) is less than α, we reject the null hypothesis. This outcome indicates that there is sufficient evidence to support the alternative hypothesis, suggesting that the frequencies of at least one of the rating categories differ significantly from the others.

In summary, the analysis reveals that the claimed distribution of customer satisfaction responses does not fit the observed data, leading to the conclusion that the responses are not uniformly distributed across the five categories.

Goodness of Fit Test: Unequal Probabilities

In statistical analysis, the goodness of fit test is a crucial method used to determine whether observed data aligns with expected distributions based on a specific claim. This test is particularly useful when dealing with distributions where probabilities are not equal, as is the case with Benford's Law. Benford's Law states that in many real-world datasets, lower digits (like 1, 2, and 3) appear more frequently than higher digits (like 7, 8, and 9).

To conduct a goodness of fit test under these conditions, the calculation of expected frequencies differs from scenarios where probabilities are equal. Instead of simply dividing the total sample size by the number of categories, expected frequencies are calculated by multiplying the total sample size by the probability of each category. For instance, if the sample size (n) is 100 and the probability of a digit appearing is 0.301, the expected frequency (E) for that digit would be:

$$E = n \times P = 100 \times 0.301 = 30.1$$

Once the expected frequencies are determined, the chi-squared statistic can be calculated using the formula:

$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

where O represents the observed frequencies. This formula quantifies the discrepancy between observed and expected frequencies, allowing for a statistical assessment of fit. The larger the difference between observed and expected values, the greater the contribution to the chi-squared statistic.

After calculating the chi-squared values for each category, these values are summed to obtain the overall chi-squared statistic. For example, if the calculated chi-squared values for various categories yield a total of 17.92, this statistic can then be compared against a chi-squared distribution table to determine the p-value and assess whether to reject or fail to reject the null hypothesis.

Understanding how to calculate expected frequencies with unequal probabilities is essential for accurately performing goodness of fit tests, particularly in real-world applications where distributions may not conform to equal probability assumptions.

A marketing associate for a supermarket chain wants to determine how many of each snack type to stock. According to previous market research, customers' preferences tend to follow the distribution in the table. If approximately 200 snack items are purchased in a day, what is the expected frequency of each snack type?

• A. 18, 11, 6, 8, 12
• B. 36, 21, 12, 8, 23
• C. 40, 40, 40, 40, 40
• D. 72, 42, 24, 16, 46

Answer: (D)

Here’s what students ask on this topic:

What is the purpose of a goodness of fit test in statistics?

A goodness of fit test is used to determine whether observed frequencies in a dataset align with expected frequencies based on a claimed distribution. It helps evaluate if the data fits a specific theoretical model or distribution. For example, you might use it to test if a die is fair by comparing the observed outcomes of rolls to the expected uniform distribution. The null hypothesis assumes the observed frequencies match the expected frequencies, while the alternative hypothesis suggests they do not. The test statistic, chi-squared (χ²), measures the discrepancy between observed and expected values, and a significant p-value indicates that the observed data does not conform to the expected distribution.

How is the chi-squared test statistic calculated in a goodness of fit test?

The chi-squared test statistic (χ²) in a goodness of fit test is calculated using the formula:

${\chi }^2=\sum _i^{}\frac{(O-E)2^{}}{E}$

Here, O represents the observed frequency, E represents the expected frequency, and the summation is performed across all categories. The formula quantifies the discrepancy between observed and expected values. Larger differences result in higher χ² values, indicating a poorer fit. Degrees of freedom, calculated as the number of categories minus one, are used to interpret the χ² value and determine the p-value, which helps decide whether to reject the null hypothesis.

What are the assumptions for conducting a goodness of fit test?

To conduct a goodness of fit test, several assumptions must be met:

• Random Sampling: The data should be collected from a random sample to ensure unbiased results.
• Observed Frequencies: Each category must have observed frequencies; no category should be empty.
• Expected Frequencies: The expected frequency for each category should be at least 5 to ensure the validity of the chi-squared approximation.

These assumptions ensure the test's reliability and accuracy. If any of these conditions are violated, the results of the test may not be valid, and alternative methods should be considered.

How do you interpret the p-value in a goodness of fit test?

The p-value in a goodness of fit test indicates the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level (α, typically 0.05), you reject the null hypothesis, concluding that the observed frequencies do not match the expected distribution. Conversely, if the p-value is greater than α, you fail to reject the null hypothesis, suggesting the observed data aligns with the expected distribution. A low p-value implies a poor fit, while a high p-value suggests a good fit.

What is the role of degrees of freedom in a goodness of fit test?

Degrees of freedom (df) in a goodness of fit test are calculated as the number of categories (k) minus one:

$\mathrm{df}=k-1$

They determine the shape of the chi-squared distribution used to calculate the p-value. Higher degrees of freedom result in a broader distribution. Degrees of freedom are essential for interpreting the test statistic and finding the corresponding p-value, which helps decide whether to reject the null hypothesis.

How do you calculate expected frequencies when probabilities are unequal?

When probabilities are unequal, expected frequencies are calculated using the formula:

$E=n\times P$

Here, n is the total sample size, and P is the probability of each category. Multiply the sample size by the probability for each category to find the expected frequency. For example, if the sample size is 100 and the probability for a category is 0.3, the expected frequency for that category is 30. This method ensures accurate calculations when probabilities vary across categories.