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. Ha: $\mu<141$ $@ \alpha=0.20$. b. $\quad H 0: \mu=-54$ vs. Ha: $\mu<-54 @ \alpha=0.05$. 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. Ha: $\mu>3.8 @ \alpha=0.001$.

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. Ha: $\mu<141$ $@ \alpha=0.20$.
b. $\quad H 0: \mu=-54$ vs. Ha: $\mu<-54 @ \alpha=0.05$.
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. Ha: $\mu>3.8 @ \alpha=0.001$.

Key Concepts

Hypothesis Testing

Hypothesis testing is a process used in statistics to evaluate evidence provided by sample data against a stated hypothesis about a population parameter. It involves establishing a null hypothesis (no effect or status quo) and an alternative hypothesis (the claim to be tested), and then using sample evidence to decide whether there is enough support to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) is the assumption that there is no effect or no difference, serving as the baseline for statistical testing. The alternative hypothesis (Ha) represents a claim that challenges the null hypothesis, indicating that there may be an effect or a difference. The decision to reject or fail to reject the null hypothesis is based on the evidence from the data.

Rejection Region

The rejection region is a critical area in the sampling distribution of the test statistic where, if the test statistic falls within this region, the null hypothesis is rejected. The boundaries of this region are determined by the significance level and the nature of the test (left-tailed, right-tailed, or two-tailed).

Significance Level

The significance level, denoted by ?, represents the probability of making a Type I error—rejecting the null hypothesis when it is actually true. It is used to define the cutoff points (critical values) for the rejection region in the hypothesis test.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis is directional, specifying that the parameter is either greater than or less than a certain value. This results in a rejection region that is confined to one end (either left or right) of the probability distribution.

Two-Tailed Test

A two-tailed test is used when the alternative hypothesis indicates that the parameter differs from a specific value, but does not indicate a direction. This leads to splitting the significance level equally between the two tails of the distribution, creating two rejection regions.

Standardized Test Statistic

The standardized test statistic is a numerical value derived from the sample data and adjusted to follow a standard probability distribution, such as the standard normal distribution (Z-distribution). This standardization allows for the determination of critical values and p-values, making it possible to compare the test statistic against known distribution properties.

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