Perform each of the following steps. a. State the hypotheses...

Perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Teachers' Salaries New York and Massachusetts lead the list of average teacher's salaries. The New York average is $\$ 76,409$ while teachers in Massachusetts make an average annual salary of $\$ 73,195 .$ Random samples of 45 teachers from each state yielded the following. $$ \begin{array}{lrr} & \text { Massachusetts } & \text { New York } \\ \hline \text { Sample means } & \$ 73,195 & \$ 76,409 \\ \text { Population standard deviation } & 8,200 & 7,800 \end{array} $$ At $\alpha=0.10$, is there a difference in means of the salaries?

Perform each of the following steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified.
Teachers' Salaries New York and Massachusetts lead the list of average teacher's salaries.
The New York average is $\$ 76,409$ while teachers in Massachusetts make an average annual salary of $\$ 73,195 .$ Random samples of 45 teachers from each state yielded the following.
$$
\begin{array}{lrr}
& \text { Massachusetts } & \text { New York } \\
\hline \text { Sample means } & \$ 73,195 & \$ 76,409 \\
\text { Population standard deviation } & 8,200 & 7,800
\end{array}
$$
At $\alpha=0.10$, is there a difference in means of the salaries?

Key Concepts

Test Statistic

The test statistic is a standardized value computed from sample data that measures the degree of divergence from what is expected under the null hypothesis. Its calculation often involves the difference between sample statistics and the hypothesized population parameters, scaled by the standard error.

Decision Rule

The decision rule is the guideline that specifies whether to reject or fail to reject the null hypothesis. It is based on comparing the test statistic against the critical values. If the test statistic falls in the rejection region, the null hypothesis is rejected.

Result Summary

The result summary is a concise interpretation of the hypothesis testing outcome, indicating whether there is sufficient statistical evidence to reject the null hypothesis. It relates the statistical findings back to the context of the research question or claim, explaining the practical significance of the results.

Critical Values

Critical values are the boundaries of the rejection region determined by the chosen significance level and the sampling distribution. They are used as benchmarks to decide whether the computed test statistic falls in a region that would lead to the rejection of the null hypothesis.

Alternative Hypothesis (Ha)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is a significant effect or difference. It represents the claim or assertion that the researcher is aiming to support with evidence from the data.

Null Hypothesis (H0)

The null hypothesis is a default assumption that there is no effect or no difference between groups or populations. It serves as the starting point for statistical testing and is assumed true until sufficient evidence is provided to support an alternative claim.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, computing a test statistic, and comparing this value to a critical threshold to determine whether to reject the null hypothesis.

Significance Level (?)

The significance level, denoted by ?, is the probability threshold used to decide whether to reject the null hypothesis. It quantifies the risk of a Type I error—rejecting the null hypothesis when it is actually true—and guides the determination of the critical values.

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