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. Step 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. Step 2: Determine the degrees of freedom for the t-distribution. The degrees of freedom is calculated as the sample size minus one. For a sample size of 51, the degrees of freedom is 50.
3. Step 3: Identify the level of significance \alpha for the confidence interval. An 80% confidence interval implies that \alpha = 1 - 0.80 = 0.20.
4. Step 4: Calculate \frac{\alpha}{2} to find the critical value for a two-tailed test. \frac{\alpha}{2} = \frac{0.20}{2} = 0.10.
5. Step 5: Use a t-distribution table or calculator to find the critical value t_{\frac{\alpha}{2}} for \frac{\alpha}{2} = 0.10 and 50 degrees of freedom. This value is the critical t-value needed for the confidence interval.