Consider the hypotheses shown below. Given that x overbarxequals=118118​, sigmaσequals=2626​, nequals=4646​, alphaαequals=0.050.05​, complete parts a through c below.
Upper H 0H0​: muμequals=125125
Upper H Subscript Upper AHA​: muμnot equals≠125125
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Part 1
a. State the decision rule in terms of the critical​ value(s) of the test statistic.
Reject the null hypothesis if the calculated value of the test​ statistic,
▼
is
▼
not contained within
greater than
equal to
less than
the critical​ value(s), enter your response here. ​Otherwise, do not reject the null hypothesis.
​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.)

We are conducting a hypothesis test for the population mean \(\mu\) using a two-tailed test with the following hypotheses:

- Null Hypothesis (\(H_0\)): \(\mu = 125\)
- Alternative Hypothesis (\(H_A\)): \(\mu \neq 125\)

### Given Information:
- Sample mean (\(\bar{x}\)): 118
- Population standard deviation (\(\sigma\)): 26
- Sample size (\(n\)): 46
- Significance level (\(\alpha\)): 0.05

### Step 1: State the test statistic formula
We use the **z-test** because the population standard deviation \(\sigma\) is known. The formula for the z-test statistic is:

\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 118\)
- \(\mu = 125\)
- \(\sigma = 26\)
- \(n = 46\)

### Step 2: Determine the critical value(s)
Since this is a two-tailed test at \(\alpha = 0.05\), we need to find the critical z-values for a two-tailed test with \(\alpha/2 = 0.025\) in each tail.

Using a z-table or calculator, the critical z-values for \(\alpha = 0.05\) are:
\[
z_{\text{critical}} = \pm 1.96
\]

### Step 3: Decision Rule
The decision rule is based on whether the calculated z-statistic falls within or outside the critical region.

- **Reject the null hypothesis** if the calculated z-statistic is **not contained within** the critical values, i.e., if:
  \[
  z < -1.96 \quad \text{or} \quad z > 1.96
  \]

- **Do not reject the null hypothesis** if the calculated z-statistic is within the range \(-1.96 \leq z \leq 1.96\).

Thus, the decision rule is:

- **Reject** \(H_0\) if \(z\) is **not contained within** the interval \([-1.96, 1.96]\).
- **Otherwise**, do not reject \(H_0\).

Would you like to proceed with calculating the test statistic and concluding the hypothesis test?

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Here are 5 related questions:
1. How do you calculate a z-statistic when the sample size changes?
2. What is the meaning of a two-tailed test compared to a one-tailed test?
3. How does the significance level \(\alpha\) affect the critical values?
4. Why do we use a z-test when the population standard deviation is known?
5. What happens if the sample mean is exactly equal to the population mean?

**Tip**: Always check whether the test is one-tailed or two-tailed before determining critical values.

Consider the hypotheses shown below. Given that x̄ = 118, σ = 26, n = 46, α = 0.05, complete parts a through c below.
H₀: μ = 125
Hₐ: μ ≠ 125
Part 1
a. State the decision rule in terms of the critical value(s) of the test statistic.
Reject the null hypothesis if the calculated value of the test statistic is not contained within the critical value(s). Otherwise, do not reject the null hypothesis.

Learn how to perform a hypothesis test for a population mean with a known standard deviation using a two-tailed test. Given sample data (x̄ = 118, σ = 26, n = 46, α = 0.05), calculate the z-statistic and apply the decision rule. This solution includes a step-by-step explanation of critical values and decision rules.

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