Find the rejection region (for the standardized test...

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed. a. $\quad H 0: \mu=141$ VS. $H a: \mu<141$ $@ \alpha=0.20 .$ b. $\quad H 0: \mu=-54$ VS. $H a: \mu<-54$ @ $\alpha=0.05 .$ C. $\quad H 0: \mu=98.6$ VS. $H a: \mu \neq 98.6$ $@ \alpha=0.05 .$ d. $\quad H 0: \mu=3.8$ VS. $H a: \mu>3.8$ @ $\alpha=0.001$

Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.
a. $\quad H 0: \mu=141$ VS. $H a: \mu<141$ $@ \alpha=0.20 .$
b. $\quad H 0: \mu=-54$ VS. $H a: \mu<-54$ @ $\alpha=0.05 .$
C. $\quad H 0: \mu=98.6$ VS. $H a: \mu \neq 98.6$ $@ \alpha=0.05 .$
d. $\quad H 0: \mu=3.8$ VS. $H a: \mu>3.8$ @ $\alpha=0.001$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether a sample of data provides enough evidence to reject a claimed hypothesis about a population parameter. It involves setting up a null hypothesis (a statement of no effect or status quo) and an alternative hypothesis, and then using sample data to determine which hypothesis is more consistent with the evidence.

Null and Alternative Hypotheses

The null hypothesis represents a default position that there is no effect or difference, while the alternative hypothesis represents a claim that contradicts the null hypothesis. In hypothesis testing, determining the direction of the alternative hypothesis (whether it is less than, greater than, or not equal to the hypothesized value) is essential for identifying the type of test and its corresponding rejection region.

Significance Level

The significance level, denoted by alpha (?), is the probability threshold for rejecting the null hypothesis when it is actually true. It determines the critical boundaries of the rejection region and reflects the investigator's tolerance for making a Type I error, which is the error of falsely rejecting the null hypothesis.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. It is defined based on the chosen significance level and the type of test (one-tailed or two-tailed). If the test statistic falls within this region, the null hypothesis is rejected in favor of the alternative.

One-Tailed and Two-Tailed Tests

One-tailed tests are used when the alternative hypothesis specifies a direction (either less than or greater than the null hypothesis value), whereas two-tailed tests are used when the alternative hypothesis only indicates a difference without specifying the direction. The type of test determines whether the rejection region is located on one side or both sides of the distribution.

Critical Value

A critical value is a threshold value derived from the distribution of the test statistic and the significance level. It marks the boundary of the rejection region. For one-tailed tests, a single critical value is used, while for two-tailed tests, there are two critical values corresponding to the lower and upper tails of the distribution.

Standardized Test Statistic

The standardized test statistic converts the observed sample statistic to a common scale under the assumed distribution (usually the standard normal distribution) for hypothesis testing. This standardization allows for the use of critical values and p-values from well-known probability distributions to determine the significance of the test result.

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