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In statistics, when dealing with observed data, it's crucial to determine what the expected frequencies are if the null hypothesis is true. Expected counts provide a baseline that allows comparison against actual observations. In our scenario with the New York City population data, expected counts are the number of each race you would expect to find in the housing complex, based on the city's distribution percentages.

To calculate these expected counts, you multiply the total number of sampled residents (800 in this case) by the percentage of each ethnic group in the city population. For instance, for the Hispanic population, the expected count is calculated as: \[ 0.28 \times 800 = 224 \]

Each ethnic group is treated similarly, enabling a clear measure against which to compare the actual counts from the sample. This, in turn, helps in calculating the chi-square statistic, which assesses whether deviations from expected counts are due to chance or represent a significant difference.

Hypothesis testing is a fundamental aspect of statistical analysis used to make inferences about populations. We start by stating two opposing hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis, denoted as \( H_0 \), assumes no effect or no difference in the situation being studied. In this exercise, the null hypothesis states that the ethnic distribution in the housing complex is the same as that observed in New York City's overall population.

• \( H_0 \): The distribution by race in the housing complex is consistent with the city's distribution.

The alternative hypothesis, or \( H_a \), suggests there is a difference or effect that is not due to random chance.

• \( H_a \): The distribution by race in the housing complex is different from the city's distribution.

Hypothesis testing involves calculating a test statistic, here the chi-square statistic, and then comparing it to a critical value to decide whether to reject the null hypothesis. It's a powerful method to validate if observed outcomes differ significantly from what was expected under the null hypothesis.

Degrees of freedom (df) is a concept used in statistical tests to determine the number of independent values or quantities which can vary in an analysis without breaking any constraints. In the chi-square test for goodness of fit, the degrees of freedom are calculated by taking the total number of categories and subtracting one.

This subtraction accounts for the use of the observed data to estimate parameters, ensuring that only the truly independent pieces of information remain. For example, in our exercise with five ethnic categories (Hispanic, Black, White, Asian, Other), the degrees of freedom is:

• \( df = 5 - 1 = 4 \)

This number is crucial when determining the critical value from the chi-square distribution table, which ultimately helps in making a decision about our hypotheses by comparing the calculated chi-square statistic.

The critical value in hypothesis testing is a threshold that determines the boundary for deciding whether to reject the null hypothesis. It is derived from probability distributions, based on the specified significance level \( \alpha \) and the degrees of freedom.

For a chi-square test, the critical value corresponds to the upper tail of the chi-square distribution. In our exercise, we use a significance level \( \alpha = 0.05 \). This means there's a 5% risk of concluding that a difference exists when there is none, defining our critical region for making this decision. With degrees of freedom \( df = 4 \), we refer to chi-square tables or statistical software to identify the critical value:

• For \( df = 4 \) and \( \alpha \) of \( 0.05 \), the critical value is approximately 9.488.

If the computed chi-square statistic exceeds this critical value, as it does in our example (26.06 > 9.488), this indicates a significant difference, leading to rejection of the null hypothesis that the observed and expected distributions are the same.