Find the critical value $t_{\frac{\alpha}{2}}$for an 80% confidence interval given a sample size of 51.

• A. 0.10
• B. 1.299
• C. 0.300
• D. 2.598

Answer: (B)

Verified step by step guidance

1. Understand that the critical value t_{\frac{\alpha}{2}} is used in constructing confidence intervals for the mean when the population standard deviation is unknown and the sample size is small.
2. Identify that the confidence level is 80%, which means the significance level \alpha is 0.20 (since \alpha = 1 - confidence level).
3. Calculate \frac{\alpha}{2} to find the tail probability for the t-distribution. For an 80% confidence interval, \frac{\alpha}{2} = 0.10.
4. Determine the degrees of freedom for the t-distribution, which is the sample size minus one. For a sample size of 51, the degrees of freedom is 50.
5. Use a t-distribution table or a statistical software to find the critical value t_{\frac{\alpha}{2}} for \frac{\alpha}{2} = 0.10 and 50 degrees of freedom.