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 $n > 30$.