To estimate the 80% confidence interval of the population mean effective period of action for the tranquilizer, we need to follow these steps:

1. Determine the critical value of t for an 80% confidence interval.
2. Calculate the maximum tolerated error (also known as the margin of error).
3. Compute the 80% confidence interval for the population mean.

Given:

• Mean ($\bar{x}$) = 1.5 hours
• Standard deviation (s) = 0.3 hours
• Sample size (n) = 8

1. Critical Value of t

For an 80% confidence interval with $n - 1 = 7$ degrees of freedom, we need to find the t-value for which the area in each tail is $0.1/2 = 0.05$.

Using a t-table or calculator: $t_{0.05, 7} \approx 1.415$

2. Maximum Tolerated Error

The margin of error (E) is calculated using the formula: $E = t \cdot \frac{s}{\sqrt{n}}$

Substituting the values: $E = 1.415 \cdot \frac{0.3}{\sqrt{8}}$ $E \approx 1.415 \cdot 0.106$ $E \approx 0.15 \text{ hours (rounded to 2 decimal places)}$

3. 80% Confidence Interval

The confidence interval is calculated as: $\bar{x} \pm E$ $1.5 \pm 0.15$

Thus, the 80% confidence interval for the population mean effective period of action is: $(1.35 \text{ to } 1.65)$

Answers:

1. The critical value of t in this case is $\approx 1.415$.
2. The maximum tolerated error on the population mean effective period of action is $0.15$ hours (rounded to 2 decimal places).
3. The 80% confidence interval of the population mean effective period of action is $1.35 \text{ to } 1.65$ hours (rounded to 2 decimal places).

Do you want details or have any questions?

Here are some more questions you might find useful:

1. How do you calculate the critical value for a given confidence level?
2. What is the difference between a t-distribution and a z-distribution?
3. How does the sample size affect the confidence interval?
4. Why is it important to round off the margin of error?
5. Can you explain the steps to calculate a confidence interval for proportions?

Tip: When estimating confidence intervals, always ensure to use the correct critical value based on the sample size and desired confidence level.