2.3 You are required to provide a 91.98% confidence interval for a mean, it is important that the interval be no longer than 2 cm . Assume that the population is normally distributed with the variance of 2 25 cm . What is the smallest sample size needed? [8]

To determine the smallest sample size needed for a confidence interval of a mean, given that the interval must be no longer than a specified length, we'll use the following information:

### Given Data:
- Confidence level: 91.98%
- Desired maximum length of the confidence interval: 2 cm
- Population variance (\( \sigma^2 \)): \( 25 \) cm²
- Population standard deviation (\( \sigma \)): \( \sqrt{25} = 5 \) cm

### Step 1: Determine the Z-value
The confidence level is 91.98%, which corresponds to a certain Z-value (critical value) in the standard normal distribution.

Since 91.98% confidence level corresponds to \( \alpha = 1 - 0.9198 = 0.0802 \), the area in each tail is \( \alpha/2 = 0.0401 \).

We'll find the Z-value that corresponds to the cumulative probability of \( 1 - 0.0401 = 0.9599 \).

### Step 2: Calculate the Sample Size
The formula for the confidence interval length is:

\[
\text{Length} = 2 \times Z \times \frac{\sigma}{\sqrt{n}}
\]

Given that the maximum length is 2 cm, we can rearrange the formula to solve for \( n \) (sample size):

\[
2 = 2 \times Z \times \frac{5}{\sqrt{n}}
\]

Simplifying, we get:

\[
1 = Z \times \frac{5}{\sqrt{n}}
\]

\[
\sqrt{n} = 5Z
\]

\[
n = (5Z)^2
\]

### Step 3: Calculate the Z-value and Substitute
We'll calculate the Z-value and then find the smallest \( n \).

I'll perform these calculations now.The smallest sample size needed is approximately **77** (since the sample size must be a whole number, we round up to the nearest integer).

### Conclusion
To ensure that the 91.98% confidence interval for the mean is no longer than 2 cm, the smallest sample size required is 77 participants.

---

Do you have any questions or need further clarification?

Here are some related questions to consider:
1. How does increasing the confidence level affect the required sample size?
2. What would happen if the population variance was different?
3. How is the sample size calculation affected if the interval length requirement is stricter?
4. Why is it necessary to round up the sample size in such calculations?
5. What is the interpretation of the confidence interval in practical terms?

**Tip:** When calculating sample sizes, always ensure that the resulting number is rounded up to guarantee that the confidence interval requirement is met.

Learn how to calculate the smallest sample size needed to achieve a 91.98% confidence interval with a maximum length of 2 cm. This problem involves statistical concepts such as confidence intervals, normal distribution, and sample size determination. Suitable for advanced undergraduate students and researchers.

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