Perform the following steps. a. State the hypotheses and...

Perform the following steps. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume all assumptions are valid. To test the effectiveness of a new drug, a researcher gives one group of randomly selected individuals the new drug and another group of randomly selected individuals a placebo. The results of the study are shown here. At $\alpha=0.10,$ can the researcher conclude that the drug results differ from those of the placebo? Use the $P$ -value method. $$ \begin{array}{lcc} \text { Medication } & \text { Effective } & \text { Not effective } \\ \hline \text { Drug } & 32 & 9 \\ \text { Placebo } & 12 & 18 \end{array} $$

Perform the following steps.
a. State the hypotheses and identify the claim.
b. Find the critical value.
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified. Assume all assumptions are valid.
To test the effectiveness of a new drug, a researcher gives one group of randomly selected individuals the new drug and another group of randomly selected individuals a placebo. The results of the study are shown here. At $\alpha=0.10,$ can the researcher conclude that the drug results differ from those of the placebo? Use the $P$ -value method.
$$
\begin{array}{lcc}
\text { Medication } & \text { Effective } & \text { Not effective } \\
\hline \text { Drug } & 32 & 9 \\
\text { Placebo } & 12 & 18
\end{array}
$$

Key Concepts

Hypothesis Testing

This is a systematic procedure for making decisions or inferences about populations based on sample data. It involves formulating a default assumption (the null hypothesis) and an alternative hypothesis, then using statistical methods to determine which hypothesis is better supported by the data.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H?) represents a statement of no effect or no difference, whereas the alternative hypothesis (H?) expresses a claim that a significant difference or effect exists. Formulating clear and testable hypotheses is the first critical step in any hypothesis testing framework.

Significance Level (?)

The significance level is a threshold set by the researcher which defines the probability of making a Type I error (rejecting the null hypothesis when it is true). It is used to determine the critical value or compare against the p-value when making a decision in a test.

Test Statistic

A test statistic is a numerical value calculated from the sample data that summarizes the evidence against the null hypothesis. Its distribution under the null hypothesis is used to determine the probability associated with the observed sample results.

P-value

The p-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It provides a measure of the strength of evidence against the null hypothesis and is used to decide whether to reject H?.

Critical Value

The critical value marks the boundary or threshold in the sampling distribution of the test statistic, beyond which the null hypothesis would be rejected. It is determined based on the significance level and the specific distribution of the test statistic.

Decision Rule

The decision rule involves comparing the p-value to the significance level (or the test statistic to the critical value) to decide whether to reject or fail to reject the null hypothesis. This rule ensures that the rejection of H? is consistent with the pre-established level of risk of error.

Conclusion Summary

After applying the decision rule, the conclusion summarizes the results of the hypothesis test in the context of the research question. It clearly states whether the evidence is sufficient to reject the null hypothesis and what that implies regarding the effectiveness or difference being tested.

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