Let's solve these two problems:

Question 17:

The probability of a caller hanging up is 40% ($P(H) = 0.4$). We are asked for the probability that one or more callers will hang up out of the next three.

The probability of at least one caller hanging up can be calculated using the complement rule: $P(\text{at least one hangs up}) = 1 - P(\text{none hang up})$

1. The probability of no one hanging up is: $P(\text{none hang up}) = (1 - 0.4)^3 = 0.6^3 = 0.216$

2. Therefore: $P(\text{at least one hangs up}) = 1 - 0.216 = 0.784$

Thus, the correct answer is 0.784.

Question 18:

We are constructing a 99% confidence interval for the mean age of male nontraditional students. The data provided includes:

• Sample mean ($\bar{x}$) = 43.75
• Sample standard deviation ($s$) = 15.05
• Sample size ($n$) = 8
• Confidence level = 99%

1. Find the critical $t^*$-value for a 99% confidence level with $n-1 = 7$ degrees of freedom: Using a $t$-table or calculator, the critical $t^*$-value is approximately $t^* = 3.499$.

2. Compute the margin of error (ME): $ME = t^* \cdot \frac{s}{\sqrt{n}} = 3.499 \cdot \frac{15.05}{\sqrt{8}} = 3.499 \cdot 5.324 = 18.65$

3. Calculate the confidence interval: $\text{CI} = \bar{x} \pm ME = 43.75 \pm 18.65$

This results in: $\text{CI} = (25.10, 62.40)$

Thus, the 99% confidence interval is (25.10, 62.40).

If you'd like, I can provide further explanation or calculations. Let me know!

Follow-up Questions:

1. Why is the complement rule useful in calculating probabilities involving "at least one" events?
2. How is the critical $t^*$-value determined for confidence intervals?
3. What assumptions must hold for the confidence interval calculation to be valid?
4. How would the interval change if the confidence level was reduced to 95%?
5. Can the process for determining $P(\text{at least one hangs up})$ be generalized for any number of callers?

Tip: When solving probability questions, always check if the complement rule simplifies the calculation—it often reduces complexity.