Estimate the 80% confidence interval of the population mean effective period of action for a certain tranquilizer which gave the following results when administered to eight patients:

Mean: 1.5 hours

Standard deviation: 0.3 hours

1. The critical value of t in this case is =

2. The maximum tolerated error on the population mean effective period of action (rounded off to 2 decimal places) is =

3. The 95% confidence interval of the population mean effective period of action (rounded off to 2 decimal places) and written in the format a to b is =

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.

Learn how to estimate the 80% confidence interval for the population mean effective period of action of a tranquilizer based on sample data. Includes calculations for critical t-value, margin of error, and confidence interval. Suitable for undergraduate students and researchers.

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