Math / Data & Statistics

QuestionFind the critical value(s) and rejection region(s) for the type of z-test with level of significance $\alpha$ . Include a graph with your answer. Two-tailed test, $\alpha=0.02$
The critical value(s) is/are $z=-2.33$ (Round to two decimal places as needed. Use a comma to separate answers as needed.)

Studdy Solution

STEP 1

1. We are conducting a two-tailed z-test.
2. The level of significance $\alpha$ is 0.02.
3. The critical values correspond to the z-scores that separate the rejection region from the acceptance region in a standard normal distribution.

STEP 2

1. Determine the critical z-values for a two-tailed test.
2. Identify the rejection regions based on the critical z-values.
3. Illustrate the rejection regions on a graph of the standard normal distribution.

STEP 3

For a two-tailed test with $\alpha = 0.02$ , the total area in the tails is 0.02. Since it is two-tailed, each tail has an area of $\frac{0.02}{2} = 0.01$ .

STEP 4

Find the critical z-values that correspond to the cumulative probabilities of 0.01 and 0.99 (since 1 - 0.01 = 0.99).
Using a standard normal distribution table or calculator: - The z-value for 0.01 is approximately $-2.33$ . - The z-value for 0.99 is approximately $2.33$ .
Critical values are: $z = -2.33, 2.33$ .

STEP 5

The rejection regions are: - Left tail: $z < -2.33$ - Right tail: $z > 2.33$

STEP 6

Illustrate the rejection regions on a graph of the standard normal distribution:
\begin{tikzpicture} \begin{axis}[ no markers, domain=-4:4, samples=100, axis lines*=left, xlabel=z $, ylabel=$ f(z), height=6cm, width=12cm, xtick={-2.33, 2.33}, ytick=\empty, enlargelimits=false, clip=false, axis on top, grid = major ] \addplot [fill=cyan!20, draw=none, domain=-4:-2.33] {gauss(0,1)} \closedcycle; \addplot [fill=cyan!20, draw=none, domain=2.33:4] {gauss(0,1)} \closedcycle; \addplot [very thick,cyan!50!black] {gauss(0,1)}; \end{axis} \end{tikzpicture}
The critical values are $z = -2.33, 2.33$ , and the rejection regions are $z < -2.33$ and $z > 2.33$ .

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Studdy Solution

STEP 1

1. We are conducting a two-tailed z-test.
2. The level of significance $\alpha$ is 0.02.
3. The critical values correspond to the z-scores that separate the rejection region from the acceptance region in a standard normal distribution.

STEP 2

1. Determine the critical z-values for a two-tailed test.
2. Identify the rejection regions based on the critical z-values.
3. Illustrate the rejection regions on a graph of the standard normal distribution.

STEP 3

For a two-tailed test with $\alpha = 0.02$ , the total area in the tails is 0.02. Since it is two-tailed, each tail has an area of $\frac{0.02}{2} = 0.01$ .

STEP 4

STEP 5

The rejection regions are: - Left tail: $z < -2.33$ - Right tail: $z > 2.33$

STEP 6

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Studdy Solution

STEP 1

1. We are conducting a two-tailed z-test.2. The level of significance α\alphaα is 0.02.3. The critical values correspond to the z-scores that separate the rejection region from the acceptance region in a standard normal distribution.

STEP 2

1. Determine the critical z-values for a two-tailed test.2. Identify the rejection regions based on the critical z-values.3. Illustrate the rejection regions on a graph of the standard normal distribution.

STEP 3

For a two-tailed test with α=0.02\alpha = 0.02α=0.02, the total area in the tails is 0.02. Since it is two-tailed, each tail has an area of 0.022=0.01\frac{0.02}{2} = 0.0120.02​=0.01.

STEP 4

STEP 5

The rejection regions are: - Left tail: z<−2.33z < -2.33z<−2.33 - Right tail: z>2.33z > 2.33z>2.33

STEP 6

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are conducting a two-tailed z-test.
2. The level of significance $\alpha$ is 0.02.
3. The critical values correspond to the z-scores that separate the rejection region from the acceptance region in a standard normal distribution.

1. Determine the critical z-values for a two-tailed test.
2. Identify the rejection regions based on the critical z-values.
3. Illustrate the rejection regions on a graph of the standard normal distribution.

For a two-tailed test with $\alpha = 0.02$ , the total area in the tails is 0.02. Since it is two-tailed, each tail has an area of $\frac{0.02}{2} = 0.01$ .

The rejection regions are: - Left tail: $z < -2.33$ - Right tail: $z > 2.33$