x = 54 n = 41 σ = 5.6 H₀ : μ = 56 Ha : μ < 56 α  = 0.05

We are tasked with performing a hypothesis test based on the information given:

- **Sample mean** (\( \bar{x} \)) = 54
- **Sample size** (\( n \)) = 41
- **Population standard deviation** (\( \sigma \)) = 5.6
- **Null hypothesis** (\( H_0 \)): \( \mu = 56 \)
- **Alternative hypothesis** (\( H_a \)): \( \mu < 56 \)
- **Significance level** (\( \alpha \)) = 0.05

### Steps for the hypothesis test:

1. **Identify the type of test:**
   - Since \( H_a \) is \( \mu < 56 \), this is a **left-tailed z-test** for the population mean.

2. **Calculate the z-test statistic:**
   The z-test statistic is given by:

   \[
   z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
   \]
   Substituting the values:

   \[
   z = \frac{54 - 56}{\frac{5.6}{\sqrt{41}}} = \frac{-2}{\frac{5.6}{6.403}} = \frac{-2}{0.8748} \approx -2.29
   \]

3. **Determine the critical value:**
   For a significance level of \( \alpha = 0.05 \), the critical value for a left-tailed z-test is approximately \( -1.645 \).

4. **Make a decision:**
   - If the calculated z-value is less than the critical value, we reject the null hypothesis.
   - In this case, \( z = -2.29 \) is less than \( -1.645 \), so we **reject the null hypothesis**.

### Conclusion:
There is sufficient evidence to reject \( H_0 \), meaning that the sample data supports the claim that \( \mu < 56 \) at the 0.05 significance level.

---

Would you like more details or explanations on any step?

Here are 5 related questions you might want to explore:

1. How is the z-test statistic calculated when the population standard deviation is unknown?
2. What is the difference between a one-tailed and a two-tailed test?
3. What happens if the significance level (\( \alpha \)) is changed?
4. How do we interpret a Type I error in this hypothesis test?
5. How can we calculate the p-value for this z-test?

**Tip:** When performing hypothesis tests, always make sure to verify whether the test should be one-tailed or two-tailed to ensure proper interpretation.

x = 54, n = 41, σ = 5.6, H₀ : μ = 56, Ha : μ < 56, α = 0.05

Explore a hypothesis test with a sample mean of 54, sample size 41, and population standard deviation of 5.6, where we test if the population mean is less than 56 using a left-tailed Z-test. This example includes detailed steps for calculating the Z-statistic and determining the critical value.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation