For Exercises I through 25, perform each of the following...

For Exercises I through 25, 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 diagrams to show the critical region (or regions), and use the traditional method of hypothesis testing unless otherwise specified. Sick Days A manager states that in his factory, the average number of days per year missed by the employees due to illness is less than the national average of $10 .$ The following data show the number of days missed by 40 randomly selected employees last year. Is there sufficient evidence to believe the manager's statement at $\alpha=0.05 ? \sigma=3.63 .$ Use the $P$ -value method. $$ \begin{array}{cccccccc}{0} & {6} & {12} & {3} & {3} & {5} & {4} & {1} \\ {3} & {9} & {6} & {0} & {7} & {6} & {3} & {4} \\ {7} & {4} & {7} & {1} & {0} & {8} & {12} & {3} \\ {2} & {5} & {10} & {5} & {15} & {3} & {2} & {5} \\ {3} & {11} & {8} & {2} & {2} & {4} & {1} & {9}\end{array} $$

For Exercises I through 25, 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 diagrams to show the critical region (or regions), and use the traditional method of hypothesis testing unless otherwise specified.
Sick Days A manager states that in his factory, the average number of days per year missed by the employees due to illness is less than the national average of $10 .$ The following data show the number
of days missed by 40 randomly selected employees last year. Is there sufficient evidence to believe the manager's statement at $\alpha=0.05 ? \sigma=3.63 .$ Use the $P$ -value method.
$$
\begin{array}{cccccccc}{0} & {6} & {12} & {3} & {3} & {5} & {4} & {1} \\ {3} & {9} & {6} & {0} & {7} & {6} & {3} & {4} \\ {7} & {4} & {7} & {1} & {0} & {8} & {12} & {3} \\ {2} & {5} & {10} & {5} & {15} & {3} & {2} & {5} \\ {3} & {11} & {8} & {2} & {2} & {4} & {1} & {9}\end{array}
$$

Key Concepts

p-value Method

The p-value method involves calculating the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. This probability is compared to the significance level. If the p-value is less than or equal to the significance level, there is sufficient evidence to reject the null hypothesis, indicating support for the alternative hypothesis.

Test Statistic

The test statistic is a standardized value calculated from sample data that measures the degree of divergence from the null hypothesis. It incorporates factors such as the sample mean, standard deviation (or standard error), and sample size. The value of the test statistic is then compared to critical values or used to compute the p-value to make a formal decision in the hypothesis test.

Decision Making in Hypothesis Testing

This involves comparing the test statistic or corresponding p-value to pre-specified thresholds (critical value or significance level) to decide whether to reject the null hypothesis. The final decision process ensures that the conclusions drawn from the hypothesis test are backed by statistically validated evidence, ultimately supporting or refuting the claim under investigation.

Critical Value

A critical value is the threshold that defines the boundary of the rejection region in hypothesis testing. It is determined based on the significance level and the distribution of the test statistic. If the computed test statistic falls beyond the critical value in the rejection region, the null hypothesis is rejected.

Null and Alternative Hypotheses

The null hypothesis (H?) represents the default or status quo assumption about a population parameter, while the alternative hypothesis (H?) represents the claim that contradicts the null. Defining these hypotheses clearly is crucial, as it determines the direction of the test (one-tailed or two-tailed) and guides the selection of the appropriate test statistic and decision criteria.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to evaluate assumptions (hypotheses) about a population parameter based on sample data. It involves setting up a null hypothesis, which represents a baseline claim, and an alternative hypothesis, which is often the claim of interest. The process uses sample data to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

Significance Level (?)

The significance level, denoted by alpha (?), is the probability threshold used to determine whether the observed sample results are sufficiently extreme to reject the null hypothesis. It quantifies the risk of a Type I error, which is the error of rejecting a true null hypothesis. Common values include 0.05 or 0.01 and are set before data analysis.

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