91Ó°ÊÓ

In statistical hypothesis testing, a null hypothesis is a kind of assumption. It suggests that there is no effect or no difference beyond the natural variation we may observe in the data. When comparing two groups, as in this exercise examining concerns about privacy among different racial groups, the null hypothesis (\( H_0 \)) assumes that the proportions of these groups are the same.

• For the current scenario, the null hypothesis is expressed as \( H_0: p_1 = p_2 \), indicating no difference in concern levels.
• The alternative hypothesis (\( H_a \)) suggests a difference exists, expressed as \( H_a: p_1 eq p_2 \).

Understanding this is critical because rejecting the null hypothesis, like in this exercise, implies there could be genuine differences in the population proportions that we have observed. The decision to reject or not reject the null hypothesis depends on the results of the statistical test performed.

The sample proportion is a statistic that estimates the proportion of individuals in a population that possess a particular characteristic. In the context of this exercise, we determine the proportion of people within each racial group who expressed concern over personal data privacy.
To find these proportions, you divide the number of concerned individuals by the total number surveyed in each group:

• For White non-Hispanics: 1617 are concerned out of 2887 surveyed, leading to a sample proportion (\( \hat{p}_1 \)) of approximately 0.560.
• For Black non-Hispanics: 325 are concerned out of 445 surveyed, yielding a sample proportion (\( \hat{p}_2 \)) of about 0.730.

Sample proportions are fundamental in hypothesis testing as they provide the actual data estimates we compare to the hypothesized proportions, under the assumption of the null hypothesis.

The standard error (SE) provides an estimate of the variability of a statistic. Specifically, it measures how much the sample proportion you calculate might vary if you took many samples.
For this exercise, we calculate the standard error of the difference between two sample proportions. This is crucial because it helps us determine how close our sample estimates might be to the true population proportions.

• To calculate the standard error: \( SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \)
• Here, \( \hat{p} \) is the pooled proportion, representing combined data from both samples, which is \(\approx 0.583\).
• The calculated standard error in this context is approximately 0.0239.

The smaller the standard error, the more precise our sample estimate is, and in hypothesis testing, it allows us to compute the Z-Statistic.

The Z-Statistic is the value we calculate to test our hypothesis. It quantifies how many standard deviations our sample statistic is away from the null hypothesis statistic.
In this case, the Z-Statistic helps determine if the difference between sample proportions is significant:

• The formula for the Z-Statistic is \( z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \)
• With our data, \( z = \frac{0.560 - 0.730}{0.0239} \approx -7.13 \)

The Z-Statistic is compared against critical values (e.g., \( \pm 1.96 \) for a 5% significance level in a two-tailed test) to decide if the null hypothesis should be rejected.
Since our calculated \( z \) is less than \(-1.96\), it leads our decision to reject the null hypothesis, indicating a statistically significant difference between the groups' proportions.