Let's carefully analyze the problem and find the smallest sample size required.

Given Data:

• Confidence level: $91.98\%$
• Desired margin of error $E$: $2 \, \text{cm}$
• Population variance $\sigma^2$: $2 \times 25 \, \text{cm} = 50 \, \text{cm}^2$
• Population standard deviation $\sigma$: $\sqrt{50} \, \text{cm}$

Step 1: Determine the z-value corresponding to the confidence level.

The confidence level is 91.98%, so the significance level $\alpha = 1 - 0.9198 = 0.0802$.

Since the confidence interval is symmetric, $\alpha/2 = 0.0401$.

Now, we find the z-value corresponding to $1 - \alpha/2 = 0.9599$.

This z-value is approximately: $z_{\alpha/2} \approx 1.75$

Step 2: Use the margin of error formula.

The margin of error $E$ for the confidence interval is given by: $E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ Given that $E = 2 \, \text{cm}$, $\sigma = \sqrt{50} \, \text{cm}$, and $z_{\alpha/2} = 1.75$, we set up the equation: $2 = 1.75 \cdot \frac{\sqrt{50}}{\sqrt{n}}$

Step 3: Solve for $n$ (sample size).

Rearrange the equation to solve for $n$: $\sqrt{n} = 1.75 \cdot \frac{\sqrt{50}}{2}$ $\sqrt{n} = \frac{1.75 \cdot 7.0711}{2} \approx \frac{12.3744}{2} \approx 6.1872$

$n \approx (6.1872)^2 \approx 38.28$

Since the sample size $n$ must be a whole number, we round up to the next whole number.

Final Answer:

The smallest sample size needed is $\boxed{39}$.

Would you like any additional details or have any questions?

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Here are five related questions you might find interesting:

1. How does the z-value change for different confidence levels?
2. How does increasing the desired margin of error affect the sample size?
3. What if the variance was $4 \times 25 \, \text{cm}^2$? How would that affect the sample size?
4. What happens to the sample size if the confidence level is increased to 95%?
5. How do we determine the sample size if the population variance is unknown?

Tip: When calculating sample size, always ensure you round up to the next whole number to maintain the desired confidence level and margin of error.