### Study on AIDS Cases by Race in Santa Clara County, California#### Reported Cases BreakdownThe table below presents the number of AIDS cases reported in Santa Clara County, categorized by race:| Race | Cases ||-----------------------|-------|| White | 28 || Hispanic | 19 || Black | 6 || Asian/Pacific Islander| 3 || **Total** | **56**|#### Chi-Square Test for Goodness-of-FitWe aim to determine if the racial distribution of these AIDS cases aligns with the overall racial distribution in Santa Clara County.**Formulate Hypotheses:**- Null Hypothesis (\(H_0\)): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.- Alternative Hypothesis (\(H_a\)): The distribution of AIDS cases is different from the race distribution in Santa Clara County.The correct null hypothesis, as indicated, is:- \(H_0\): The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.#### Population Distribution by RaceTo calculate the expected number of AIDS cases for each racial group, use the population distribution of Santa Clara County:| Race | Proportion | Expected Cases (Fill in) ||-----------------------|------------|--------------------------|| White | 0.441 | || Hispanic | 0.25 | || Black | 0.028 | || Asian/Pacific Islander| 0.281 | || **Total** | **1** | |Use these proportions to compute expected cases, rounding your answers to three decimal places if necessary.

Given: Race Cases White 28 Hispanic 19 Black 6 Asian/Pacific Islander 3 Total 56 State the hypotheses as follows: H0:The distribution of AIDS cases is consistent with the race distribution in Santa Clara County. Ha:The distribution of AIDS cases is different from the race distribution in Santa Clara County. The expected cases are obtained as follows: Race Proportion Expected cases White 0.441 56×0.441=24.696 Hispanic 0.25 56×0.25=14 Black 0.028 56×0.028=1.568 Asian/Pacific Islander 0.281 56×0.281=15.736 Total 1 56 The Chi-square test statistic is computed as follows: χ2=∑Oi-Ei2Ei=28-24.696224.696+19-14214+6-1.56821.568+3-15.736215.736=0.442031746+1.785714286+12.52718367+10.30793696=25.06286666≈25.063 Degrees of freedom: df=k-1=4-1=3 Use Excel function "=CHISQ.DIST.RT(25.063,3)", the p-value obtained is 0.000015 Since the p-value is less than significance level, reject the null hypothesis. Therefore, the differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County is statistically significant. The differences are also important from a practical perspective.