Chapter 9: Q. 9.82 (page 380)

In the given exercise, we have provided a sample mean, sample size, and population standard. In each case, use the one-mean z-test to perform the required hypothesis test at the 5% significance level.
$\overset{—}{x} = 20, n = 24, σ = 4, H_{0} : μ = 22, H_{a} : μ \neq 22$

Short Answer

Expert verified

Ans: We have to reject $H_{0} a t 5 %$ the level of significance as the value of z falls in the rejection region.

Step by step solution

Step 1. Given information.

given,

$\overset{—}{x} = 20, n = 24, σ = 4, H_{0} : μ = 22, H_{a} : μ \neq 22$

Step 2. Let's perform a hypothesis test.

$\begin{aligned} H_{0} : μ = 22 \\ H_{a} : μ \neq 22 \end{aligned}$

Where $μ$ is the population mean.

Since the alternative hypothesis contains the symbol localid="1651574793371" $\neq$ .

And the test is a two-tailed test.

Now,

The level of significance is, $α = 0.05$ .

Step 3. Now, we find the hypothesis test about the mean, μ.

we have,

$\overset{—}{x} = 20, n = 24, σ = 4$

Test statistic

$\begin{aligned} z = \frac{\bar{x} - μ_{0}}{\frac{σ}{\sqrt{n}}} \\ = \frac{20 - 22}{\frac{4}{\sqrt{24}}} \\ = - 2.45 \end{aligned}$

Step 4. Since this is a two-tailed test with α=0.05,

By using the normal distribution table, the critical values are:

$\begin{aligned} \pm z_{α / 2} = \pm z_{0.05 / 2} \\ = \pm z_{0.025} \\ = \pm 1.96 \end{aligned}$

Here the rejection regions are $z < - z_{α / 2} o r z > z_{α / 2}$

i.e., $z < - 1.96 o r z > 1.96$

Here $z = - 2.45 < - z_{0.025} = - 1.96$

Since the level of significance as the value of z falls in the rejection region.

So, we have to reject localid="1651211012259" $H_{0} a t 5 %$ .

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In the given exercise, we have provided a sample mean, sample size, and population standard. In each case, use the one-mean z-test to perform the required hypothesis test at the 5% significance level.
$\overset{—}{x} = 20, n = 24, σ = 4, H_{0} : μ = 22, H_{a} : μ \neq 22$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Let's perform a hypothesis test.

Step 3. Now, we find the hypothesis test about the mean, μ.

Step 4. Since this is a two-tailed test with α=0.05,

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In the given exercise, we have provided a sample mean, sample size, and population standard. In each case, use the one-mean z-test to perform the required hypothesis test at the 5% significance level.x—=20,n=24,σ=4,H0:μ=22,Ha:μ≠22

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Let's perform a hypothesis test.

Step 3. Now, we find the hypothesis test about the mean, μ.

Step 4. Since this is a two-tailed test with α=0.05,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

In the given exercise, we have provided a sample mean, sample size, and population standard. In each case, use the one-mean z-test to perform the required hypothesis test at the 5% significance level.
$\overset{—}{x} = 20, n = 24, σ = 4, H_{0} : μ = 22, H_{a} : μ \neq 22$