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. In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At $\alpha=0.05,$ can it be concluded that there is a difference in the selfesteem scores of the two groups? Use the $P$ -value method. $$ \begin{array}{ll} \text { Leavers } & \text { Stayers } \\ \hline \bar{X}_{1}=3.05 & \bar{X}_{2}=2.96 \\ \sigma_{1}=0.75 & \sigma_{2}=0.75 \\ n_{1}=103 & n_{2}=225 \end{array} $$

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.
In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At $\alpha=0.05,$ can it be concluded that there is a difference in the selfesteem scores of the two groups? Use the $P$ -value method.
$$
\begin{array}{ll}
\text { Leavers } & \text { Stayers } \\
\hline \bar{X}_{1}=3.05 & \bar{X}_{2}=2.96 \\
\sigma_{1}=0.75 & \sigma_{2}=0.75 \\
n_{1}=103 & n_{2}=225
\end{array}
$$

Key Concepts

Decision Rule and Conclusion

This involves comparing the test statistic or p-value with the pre-determined critical value or significance level. If the p-value is less than the significance level (or if the test statistic falls in the rejection region), the null hypothesis is rejected. The conclusion summarizes the results in context, stating whether there is statistically significant evidence to support the existence of a difference.

P-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the observed value. It provides a measure of the evidence against the null hypothesis and is compared to the significance level to decide on its rejection.

Critical Values

Critical values are the cutoff points in the sampling distribution of the test statistic beyond which the null hypothesis is rejected. They are based on the significance level and the distribution (such as the normal or t-distribution) and define rejection regions for the hypothesis test.

Test Statistic

The test statistic is a standardized value calculated from sample data that is used to compare against a theoretical distribution to decide whether to reject the null hypothesis. For comparing two means, it typically involves the difference between sample means normalized by the standard error.

Significance Level (?)

The significance level is the threshold set by the researcher, representing the probability of rejecting the null hypothesis when it is actually true (Type I error). It determines how strong the evidence must be before rejecting the null hypothesis.

Null and Alternative Hypotheses

This concept involves formally stating the assumption to be tested. The null hypothesis (H?) typically asserts that there is no difference or effect, while the alternative hypothesis (H?) states that there is a difference. These hypotheses define the framework for the test and establish what is meant by 'no difference' versus 'a difference.'

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