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

Find the rejection region (for the standardized test statistic) for each hypothesis test. a. $\quad H 0: \mu=27$ VS. $H a ; \mu<27$ $@ \alpha=0.05$. b. $\quad H 0: \mu=52$ vs. $H a: \mu \neq 52$ $@ \alpha=0.05 .$ c. $\quad H 0: \mu=-105$ VS. Ha: $\mu>-105$ $@ \alpha=0.10$. d. $\quad H 0: \mu=78.8$ VS. Ha: $\mu \neq 78.8 @ \alpha=0.10$.

Find the rejection region (for the standardized test statistic) for each hypothesis test.
a. $\quad H 0: \mu=27$ VS. $H a ; \mu<27$ $@ \alpha=0.05$.
b. $\quad H 0: \mu=52$ vs. $H a: \mu \neq 52$ $@ \alpha=0.05 .$
c. $\quad H 0: \mu=-105$ VS. Ha: $\mu>-105$ $@ \alpha=0.10$.
d. $\quad H 0: \mu=78.8$ VS. Ha: $\mu \neq 78.8 @ \alpha=0.10$.

Key Concepts

Significance Level (?)

The significance level, denoted as ?, represents the probability of committing a Type I error—that is, rejecting the null hypothesis when it is actually true. It is used to determine the critical value(s) of the test statistic that form the boundaries of the rejection region.

Two-Tailed Test

A two-tailed test is applied when the alternative hypothesis states that the parameter differs from a specified value in either direction. In this case, the rejection region is split between the two tails of the distribution, and the significance level is divided equally to account for deviations in both directions.

Standardized Test Statistic

A standardized test statistic is a measure that converts the sample statistic into a common scale, typically using a Z or t transformation, so that its distribution under the null hypothesis is known. This allows for the use of standard probability tables to assess the probability of observing the computed statistic.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction (either less than or greater than) relative to a hypothesized parameter. In such tests, the rejection region is located entirely in one tail of the standard distribution, reflecting the directional nature of the test.

Rejection Region

The rejection region is the set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined using the significance level (?) and indicates the most extreme outcomes that are considered unlikely under the null hypothesis.

Transcript

00:01 Okay, so in this problem, we are trying to determine the rejection region based on the standardized test statistic, which is our z score, for a given hypothesis test.

00:11 So the first one we're given is we're testing against a mu less than 27 with a significance level of 0 .05.

00:19 So this is a one -tailed test, and it's a one -tailed to the left because it is a less than, not a less than or equal to sign, and it's to the left.

00:30 Because it's less than, not greater than.

00:33 So then all we have to do is we need to go to our appendix in the back of the book, figure 2 -3, and go all the way down to the bottom when our area to the left is 0 .05, and then we get a z score of 1 .645.

00:52 This one is negative because we're looking at less than, so it's a negative 1 .645.

00:58 Okay.

01:00 So then part b, we are testing against mu not equal to 52 with a significance test of 0 .05.

01:08 So that's a two -teld test.

01:10 So we're looking at either or being each one.

01:18 So with that case, we're going to take our significance level and divide it by 2 because 5 % of our curve is not right here.

01:25 5 % is contained within both curves.

01:28 Okay, so with that being said, we're looking for the z score at where it's 2 .5.

01:32 Percent here and two and a half percent on the other end...

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