https://file.aiquickdraw.com/mathbot/file/173025795851128twx2s7.jpg

Let's analyze part (b) from your question. 

The given parameters are:
- Significance level, \( \alpha = 0.01 \)
- Sample size, \( n = 28 \)
- Alternative hypothesis, \( H_1 : \mu \neq \mu_0 \)

Since this is a two-tailed test (due to \( H_1 : \mu \neq \mu_0 \)), we need to divide the significance level by 2 to get the area in each tail. Thus, for each tail, the area is \( \alpha/2 = 0.01/2 = 0.005 \).

1. **Determine degrees of freedom**: The degrees of freedom (df) is \( n - 1 = 28 - 1 = 27 \).

2. **Find the critical \( t \)-value**: Using a \( t \)-table or calculator, we look for the critical \( t \)-value at 27 degrees of freedom with an area of 0.005 in the upper tail (since it’s two-tailed, we’ll use both positive and negative values of this critical \( t \)).

From the table, the critical \( t \)-value for 27 degrees of freedom and \( \alpha = 0.005 \) in one tail is approximately **2.771**.

### Answer:
The critical \( t \)-values are \( t = \pm 2.771 \) for part (b).

Would you like more details on the calculations or a deeper explanation of any step?

---

Here are some follow-up questions to extend your understanding:

1. How would the critical \( t \)-value change if the sample size were increased?
2. What would be the \( t \)-value if the significance level \( \alpha \) were set to 0.05 instead?
3. How does the critical value of \( t \) differ between a one-tailed and a two-tailed test?
4. Why do we use degrees of freedom (df) when looking up \( t \)-values?
5. In what scenarios would you use a \( z \)-distribution instead of a \( t \)-distribution?

**Tip**: Remember that the \( t \)-distribution becomes closer to the \( z \)-distribution as the sample size increases, particularly when \( n > 30 \).

Use Critical Values for the Student's t Distribution Table to find the critical value or values for the following values of the significance level α, sample size n, and alternate hypothesis H_1:

(b) α = 0.01, n = 28, and H_1 : μ ≠ μ_0

Learn how to determine critical t-values for hypothesis testing with a significance level of α = 0.01, a sample size of 28, and a two-tailed test for Student's t-distribution. Includes step-by-step calculation of degrees of freedom and critical values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation