For Exercises 2 through 12, perform each of these steps....

For Exercises 2 through $12,$ perform each of these steps. Assume that all variables are normally or approximately normally distributed. 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. Improving Study Habits As an aid for improving students' study habits, nine students were randomly selected to attend a seminar on the importance of education in life. The table shows the number of hours each student studied per week before and after the seminar. At $\alpha=0.10$, did attending the seminar increase the number of hours the students studied per week? $$ \begin{array}{l|ccccccccc}{\text { Before }} & {9} & {12} & {6} & {15} & {3} & {18} & {10} & {13} & {7} \\ \hline \text { After } & {9} & {17} & {9} & {20} & {2} & {21} & {15} & {22} & {6}\end{array} $$

For Exercises 2 through $12,$ perform each of these steps. Assume that all variables are normally or approximately normally distributed.
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.
Improving Study Habits As an aid for improving students' study habits, nine students were randomly selected to attend a seminar on the importance of education in life. The table shows the number of hours each student studied per week before and after the seminar. At $\alpha=0.10$, did attending the seminar increase the number of hours the students studied per week?
$$
\begin{array}{l|ccccccccc}{\text { Before }} & {9} & {12} & {6} & {15} & {3} & {18} & {10} & {13} & {7} \\ \hline \text { After } & {9} & {17} & {9} & {20} & {2} & {21} & {15} & {22} & {6}\end{array}
$$

Key Concepts

Decision Rule and Conclusion

The decision rule in hypothesis testing involves comparing the test statistic to the critical value(s). If the test statistic falls into the rejection region defined by the critical value, the null hypothesis is rejected. The final conclusion should be summarized in the context of the research question, indicating whether there is statistically significant evidence to support the alternative hypothesis or not.

Critical Value

The critical value in hypothesis testing is a threshold derived from the chosen significance level (?) and the relevant probability distribution (such as the t-distribution). It defines the boundary of the rejection region for the null hypothesis. If the test statistic exceeds this value in the direction specified by the alternative hypothesis, the null hypothesis is rejected.

Test Statistic

The test statistic is a standardized value calculated from the sample data that quantifies how far the sample statistic (such as the mean difference in a paired t-test) is from the value stated in the null hypothesis. It is used to determine the p-value or to compare against critical values, thereby informing the decision to reject or fail to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents a position of no change or no effect, while the alternative hypothesis (H?) represents a claim to the contrary. For example, in studies assessing before-and-after scenarios, the null hypothesis might state that there is no difference between the two measurements, and the alternative hypothesis might claim that there is an increase or decrease. The direction of the alternative hypothesis (one-tailed or two-tailed) depends on the specific claim being investigated.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves setting up a null hypothesis (a statement of no effect or no difference) and an alternative hypothesis (a statement indicating the presence of an effect or a difference), then using sample data to assess the likelihood of observing the data if the null hypothesis were true.

Paired t-test

A paired t-test is used when the data consists of paired observations, such as measurements taken before and after a treatment on the same subjects. This test assesses whether the mean difference between the paired observations is statistically different from zero, accounting for the fact that the samples are related and reducing variability compared to independent samples.

Transcript

00:01 For this question, we have a random sample of nine students for the average number of hours they studied per week before attending a seminar in education and after attending the seminar.

00:12 And we're interested to know if there's evidence to suggest that the seminar increased the hours that students were studying.

00:20 And we're asked to test this at the alpha equals 0 .1 level.

00:28 So the first step is to identify the hypotheses.

00:31 So the no hypothesis is that the mean difference is zero, or that there was no change.

00:44 And the alternative hypothesis is that the mean difference in hours per week is less than zero.

00:54 That would indicate that the students after the seminar on average are studying more.

01:01 Now the next step is to calculate the critical value, so we have alpha equals 0 .10.

01:07 Also we have the sample size is 9 from the question.

01:12 So that gives us a degrees of freedom of eight.

01:17 So if we go to the t table and look this up, so we have eight degrees of freedom and an area of 0 .1 in one tail, because this is a left -tailed test.

01:38 And so our critical value is 1 .397.

01:48 And again, this is a left -tailed test, which means that it's negative 1 .397.

01:56 We know it's a left -tailed test because of our alternative hypothesis is less than zero.

02:04 And so our test value is of this form...

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