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. Find the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed. A person read that the average number of hours an adult sleeps on Friday night to Saturday morning was 7.2 hours. The researcher feels that college students do not sleep 7.2 hours on average. The researcher randomly selected 15 students and found that on average they slept 8.3 hours. The standard deviation of the sample is 1.2 hours. At $\alpha=0.05,$ is there enough evidence to say that college students do not sleep 7.2 hours on average?

Perform each of the following steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Find the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed.
A person read that the average number of hours an adult sleeps on Friday night to Saturday morning was 7.2 hours. The researcher feels that college students do not sleep 7.2 hours on average. The researcher randomly selected 15 students and found that on average they slept 8.3 hours. The standard deviation of the sample is 1.2 hours. At $\alpha=0.05,$ is there enough evidence to say that college students do not sleep 7.2 hours on average?

Key Concepts

Result Interpretation

After making a decision based on the test statistic and critical values, the final step is to interpret the results in context. This involves summarizing whether there is sufficient evidence to support the alternative hypothesis and explaining the practical implications of the decision relative to the research question.

Decision Rule

Once the test statistic and critical value(s) have been determined, the decision rule is applied to decide whether to reject or fail to reject the null hypothesis. If the test statistic falls in the rejection region (beyond the critical value), the null hypothesis is rejected; otherwise, there is insufficient evidence to reject it.

Critical Value(s) and Rejection Regions

Critical values define the boundaries of the rejection region(s) for the test statistic at the chosen significance level. These values are derived from the appropriate probability distribution (such as the t-distribution or normal distribution). If the computed test statistic falls into the rejection region beyond these critical values, it indicates that the observed result is unlikely under the null hypothesis.

Significance Level

The significance level, denoted as ?, is a pre-specified threshold that defines the probability of making a Type I error—that is, rejecting the null hypothesis when it is actually true. It sets the standard for how extreme the sample data must be to conclude that there is enough evidence against the null hypothesis.

Hypothesis Formulation

In any hypothesis test, you begin by stating two opposing hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents a default position or status quo (for instance, that a parameter equals a certain value), while the alternative hypothesis reflects the researcher's claim or what they suspect to be true instead. This step also involves clearly identifying which hypothesis represents the claim being tested.

Test Statistic Calculation

The test statistic is a standardized value computed from sample data that measures how far the sample estimate deviates from the value stated in the null hypothesis. This calculation often involves the sample mean, standard deviation, and sample size, and, depending on the situation (e.g., small sample size or unknown population standard deviation), it may follow a t-distribution or a normal distribution.

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