## What are expected frequencies in goodness-of-fit tests?

Expected frequencies in goodness-of-fit tests represent the frequencies that would be expected in each category if the observed data perfectly fit the expected distribution. These tests are used to assess whether the observed data significantly deviate from the expected distribution. By comparing the observed frequencies to the expected frequencies, we can determine if there is a significant difference or discrepancy that cannot be attributed to random variation.

In order to compute the expected frequencies, we first need to define the expected distribution. This distribution is usually based on a theoretical or hypothesized model that describes the expected proportions or probabilities in each category. For example, if we are conducting a goodness-of-fit test to examine the distribution of eye colors in a population, our expected distribution might be based on the hypothesis that 25% of the population has blue eyes, 40% has brown eyes, 30% has green eyes, and 5% has other colors.

Once we have the expected distribution, we can calculate the expected frequencies. This is done by multiplying the total sample size (or the total number of observations) by the expected proportion or probability for each category. In our eye color example, if we have a sample size of 1000 individuals, the expected frequency for blue eyes would be 1000 * 0.25 = 250, for brown eyes it would be 1000 * 0.40 = 400, for green eyes it would be 1000 * 0.30 = 300, and for other colors it would be 1000 * 0.05 = 50.

These expected frequencies provide a baseline against which we can compare the observed frequencies. The observed frequencies are the actual frequencies or counts obtained from the sample data. In our eye color example, if we collected data from our sample of 1000 individuals and found that 280 had blue eyes, 410 had brown eyes, 290 had green eyes, and 20 had other colors, these would be the observed frequencies.

By comparing the observed frequencies to the expected frequencies, we can determine if the differences between these two sets of frequencies are statistically significant. This is done using statistical tests such as the chi-square test. The chi-square test calculates a test statistic that measures the discrepancy between the observed and expected frequencies, and determines whether this discrepancy is unlikely to occur by chance alone.

If the test statistic is found to be statistically significant, it indicates that there is a significant difference between the observed and expected frequencies. This suggests that the observed data deviate from the expected distribution and provides evidence for rejecting the null hypothesis of perfect fit. On the other hand, if the test statistic is not statistically significant, it suggests that there is no significant difference and we fail to reject the null hypothesis, indicating that the observed data are consistent with the expected distribution.

Overall, expected frequencies play a crucial role in goodness-of-fit tests as they provide a reference point for comparing observed frequencies and assessing the significance of any deviations. These tests are widely used in various fields such as psychology, biology, sociology, and market research to examine the fit of observed data to theoretical models or expected distributions, thus helping researchers uncover patterns and relationships in their data.

## How are expected frequencies computed for goodness-of-fit tests?

Expected frequencies are computed for goodness-of-fit tests using a specified mathematical model or assumption about the distribution of the observed data. These expected frequencies represent the values that would be expected to occur if the observed data perfectly conformed to the assumed distribution. By comparing the observed frequencies with the expected frequencies, statisticians can determine whether there is a significant difference between the observed and expected data.

In order to compute the expected frequencies, the first step is to choose an appropriate distribution for the data. This distribution is typically based on prior knowledge, theoretical considerations, or expert opinion. For example, if the data represents the outcome of tossing a fair six-sided die, the expected frequencies would be evenly distributed across all six possible outcomes.

Once the distribution is selected, the expected frequencies are calculated using mathematical formulas. The specific method for computing expected frequencies varies depending on the type of distribution being used. For example, if the data follows a normal distribution, the expected frequencies can be computed based on the mean and standard deviation of the observed data.

After computing the expected frequencies, the next step is to compare them with the observed frequencies. This is typically done using the Chi-Square test, which determines the degree of deviation between the observed and expected frequencies. The Chi-Square statistic is computed by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies.

The resulting Chi-Square statistic follows a Chi-Square distribution, which has a known probability distribution. By comparing the computed Chi-Square statistic with the critical values from the Chi-Square distribution, statisticians can determine whether the deviation between the observed and expected frequencies is statistically significant.

If the computed Chi-Square statistic is greater than the critical value, it indicates that there is a significant difference between the observed and expected frequencies, suggesting that the data does not fit the assumed distribution. Conversely, if the computed Chi-Square statistic is less than the critical value, it suggests that the data is in good agreement with the assumed distribution.

In summary, expected frequencies for goodness-of-fit tests are computed by selecting an appropriate distribution, calculating the expected frequencies based on mathematical formulas, and then comparing them with the observed frequencies using the Chi-Square test. This statistical test helps to determine whether there is a significant difference between the observed and expected data, providing insights into the goodness-of-fit of the assumed distribution.