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The null hypothesis is the foundation of hypothesis testing. It represents the default or original assumption we aim to test against. In this context, the null hypothesis, denoted as \( H_0 \), claims that there is no discrimination against black individuals in grand jury selection.
Mathematically, it assumes the proportion of eligible black individuals is \( 0.25 \). This means that, according to the null hypothesis, 25% of those eligible for jury duty should be black, as predicted by census data. The null hypothesis acts as a baseline for testing whether the observed data deviates significantly from this assumption.
Rejecting the null hypothesis implies evidence contrary to this established belief. It suggests that the actual proportion might be significantly lower than 0.25 due to factors like discrimination.

The alternative hypothesis, represented as \( H_a \), offers a contrasting perspective to the null hypothesis. It proposes what we suspect might be true: that there is indeed discrimination, leading to a lower proportion of black individuals in grand jury selection compared to the stated 0.25.
In this exercise, \( H_a \) is mathematically defined as \( p < 0.25 \). This indicates our suspicion that the actual proportion is less than 25%.
If the alternative hypothesis is supported by the data, it suggests that the observed sample proportion of 0.1686 could be due to genuine discrimination rather than random variability. Thus, the alternative hypothesis directs the aim of the test to identify evidence of an effect contrary to the assumed norm in the null hypothesis.

The critical value is a threshold we use to decide whether to reject the null hypothesis. In hypothesis testing, it demarcates the boundary of the null hypothesis's rejection region within the chosen significance level, \( \alpha \).
For this scenario, with a significance level of 0.01, we employ a one-tailed test because we suspect the proportion is specifically lower than 0.25. This gives us a critical value of approximately \(-2.33\) from the standard normal distribution.
By comparing the test statistic to the critical value, we determine if our data falls within the range of unlikely outcomes under \( H_0 \). If the test statistic is lower than this critical value, it indicates that the result is significant enough to reject \( H_0 \). The critical value is essential as it helps us to make an informed decision about the validity of the null hypothesis.

The test statistic is a calculation that helps us quantify the difference between the observed data and what is expected if the null hypothesis were true. In this case, we're using a z-test for proportions, ascertaining how many standard deviations the observed sample proportion \( \hat{p} = 0.1686 \) is away from the expected proportion \( p_0 = 0.25 \).
Employing the formula \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \), where \( n = 1050 \), we calculate a test statistic of approximately \(-5.98\).
This negative value signifies that the sample proportion is significantly lower than the expected proportion. Given it exceeds the critical value \(-2.33\), it further reinforces that the data strongly contradicts \( H_0 \), supporting the presence of an effect, potentially indicating discrimination within the context of jury selection.