To find the critical value for constructing a confidence interval at a 98% confidence level, we'll use the $z$-score corresponding to 98% confidence. Since we are working with two independent samples and the standard deviations are known, we use the normal distribution $z$-score for this calculation.

Step-by-Step Solution

1. Identify the Confidence Level: We have a 98% confidence level.

2. Determine the Tail Probability: For a 98% confidence interval, there is a 2% (or 0.02) total probability in the two tails of the normal distribution (1 - 0.98 = 0.02). Since the distribution is symmetric, each tail has 1% (or 0.01) of the probability.

3. Find the Critical Value $z_{\alpha/2}$:

   • We need the $z$-score that leaves 1% in the upper tail.
   • This $z$-score can be found from a standard normal distribution table, or through a calculator.

   For a 98% confidence level, the critical $z$-value (denoted $z_{\alpha/2}$) is approximately: $z_{\alpha/2} \approx 2.33$

Answer

The critical value to use for constructing a 98% confidence interval is: $z_{\alpha/2} = 2.33$

Would you like to proceed with the next step of calculating the confidence interval?

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Here are some follow-up questions to deepen understanding:

1. Why is a $z$-score used instead of a $t$-score in this scenario?
2. How would the critical value change if we had a 95% confidence level instead?
3. What would the interpretation of a 98% confidence interval be in this context?
4. How does sample size influence the width of the confidence interval?
5. Why do we divide the tail probability by two for a two-tailed confidence interval?

Tip: For higher confidence levels, the critical value increases, making confidence intervals wider.