91Ó°ÊÓ

In hypothesis testing, the null hypothesis is a statement that there is no effect or no difference, and it serves as the starting point for statistical analysis. It represents the idea that any observed effect is due to chance.
This hypothesis is typically denoted as \(H_0\). In our exercise, the null hypothesis states that the proportion of left-handed lawyers is the same as the proportion of left-handed Americans, \(P_l = 0.10\).

• The null hypothesis provides a baseline for comparison.
• If the data shows a significant difference, the null hypothesis may be rejected.

Rejecting the null hypothesis suggests that there is enough evidence to support the alternative hypothesis.

The significance level, often denoted by \(\alpha\), is a threshold used to determine whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true, known as a Type I error.
Common significance levels are 0.05 (5%) and 0.10 (10%).

• At a 5% significance level, there is a 5% risk of concluding that a difference exists when there isn't one.
• At a 10% significance level, this risk increases to 10%.

In our exercise, we compare the p-value to the significance levels to decide whether to reject the null hypothesis. If the p-value is less than \(\alpha\), we reject \(H_0\). The study concludes that the proportion of left-handed lawyers is different from the general population at both significance levels.

The test statistic helps determine how far the observed data is from the null hypothesis. It is calculated using a formula that depends on the type of test being conducted.
For proportion tests, the Z-test statistic is used, calculated as follows: \[Z = \frac{P_l - P_0}{\sqrt{P_0(1-P_0)/n}}\]Here, \(P_l\) is the sample proportion, \(P_0\) is the population proportion, and \(n\) is the sample size. In the exercise, we find:- \(P_l = \frac{16}{105} \approx 0.152\)- \(P_0 = 0.10\)- \(n = 105\)Substituting these into the formula gives \(Z \approx 2.04\).
This Z-value is then used to find the p-value, which shows how extreme the observed data is under the null hypothesis. The smaller the p-value, the greater the evidence against \(H_0\). The exercise's p-value of 0.0414 leads to the rejection of \(H_0\) at both significance levels.