91Ó°ÊÓ

In hypothesis testing, the critical value acts as a crucial threshold. It determines whether the test statistic's result supports rejecting the null hypothesis or not. When you perform a test, you first decide on the level of significance, usually represented by \(\alpha\), to gauge the risk you're willing to take of rejecting a true null hypothesis. In this case, \(\alpha = 0.05\).
The critical value is based on the type of test. For a two-tailed test, like this one, the critical values cut off the extreme 5% of the distribution, both 2.5% in each tail. Using the standard normal distribution table, the critical values here are approximately \(+1.96\) and \(-1.96\). If your computed test statistic falls beyond these critical values, you reject the null hypothesis.

• Critical values indicate cut-off points for acceptance or rejection of hypotheses.
• Calculated using the significance level and the type of statistical test.
• In a two-tailed test with \(\alpha = 0.05\), it is typically \(\pm 1.96\).

The test statistic is a value derived from your sample data during hypothesis testing. It directly influences the decision-making process. You calculate it based on the difference between sample statistics. In this scenario, it compares the proportion of high school graduates from two states.
For this exercise, we use the formula primarily for the difference in proportions:\[z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\]Where:

• \(\hat{p}_1\) and \(\hat{p}_2\) are sample proportions for Pennsylvania and Idaho respectively,
• \(\hat{p}\) is the pooled proportion across both samples.

By substituting the values, we already computed, the test statistic \(z\) ended up as approximately \(2.05\). Since \(z\) lies outside the range of critical values, it suggests a significant difference.

The null hypothesis is a foundational concept in hypothesis testing. It represents a statement or proposition that there is no effect or no difference. In its essence, it's the default or "no change" assumption.
In this context, the null hypothesis (\(H_0\)) asserts that the graduation rates between the two states, Pennsylvania and Idaho, are the same. Mathematically, this means: \[H_0: p_1 = p_2\]Here, \(p_1\) and \(p_2\) represent the proportions of graduates from Pennsylvania and Idaho, respectively.

• The null hypothesis is generally the claim you try to find evidence against.
• A failure to reject \(H_0\) implies no significant difference was found.
• Remember, rejecting it does not confirm the alternative hypothesis completely but shows a significant sample-based evidence against it.

The alternative hypothesis challenges the null hypothesis, suggesting that there is indeed an effect or a difference. In many situations, this is the research hypothesis we're driven to support with statistical evidence.
For our problem, the alternative hypothesis (\(H_1\)) states there is a difference in the proportions of graduates between the states:\[H_1: p_1 eq p_2\]This hypothesis aligns with our initial claim and is interested in proving the disparity between graduation rates in Pennsylvania and Idaho.

• The alternative hypothesis is what you might suspect is true before collecting data.
• Success in rejecting \(H_0\) usually provides enough evidence to support \(H_1\).
• Here, it indicated that the graduation rates between the states are not equal.