When constructing a $90\%$ confidence interval for the population mean with a sample size of $n=50$, which constant should be used as the critical value if the population standard deviation is known?

• A. $1.28$
• B. $1.645$
• C. $1.96$
• D. $2.576$

Answer: (B)

Verified step by step guidance

1. Identify the confidence level given in the problem, which is 90%. This means the confidence interval should capture the true population mean 90% of the time.
2. Since the population standard deviation is known and the sample size is 50 (which is greater than 30), we use the standard normal distribution (Z-distribution) to find the critical value.
3. The critical value corresponds to the Z-score that leaves an area of \( \frac{1 - 0.90}{2} = 0.05 \) in each tail of the standard normal distribution.
4. Look up the Z-score for the cumulative probability of \( 1 - 0.05 = 0.95 \) in the standard normal table or use a statistical software to find the critical value.
5. The critical value you find from the Z-table for 90% confidence level is the constant to use in the confidence interval formula: \( \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \).