Question

Find the critical value tα/2 needed to construct a confidence interval of the given level with the given sample size:

a) Level 99%, sample size 6
b) Level 99.9, sample size 16
c) Level 99.8%, sample size 10

Answer 1

Critical t values are found by determining the degrees of freedom (sample size - 1), finding α/2 based on the confidence level, and using a t-distribution table or calculator function to find the critical value tα/2 for the determined degrees of freedom and α/2.

Step-by-step explanation:

The task is to find the critical value tα/2 needed to construct a confidence interval of the given level with the given sample size for three different scenarios: a) Level 99%, sample size 6 b) Level 99.9, sample size 16 c) Level 99.8%, sample size 10.

Here is how to find the critical t values:

1. Determine the degrees of freedom (df), which is the sample size minus one.
2. Since the confidence interval is two-tailed, divide the desired level of confidence's complement (1 - confidence level) by 2 to get α/2.
3. Use a t-distribution table or a calculator function, like invT, to find the critical value tα/2 for the calculated df and α/2.
4. For each scenario:

• a) For a 99% confidence level with a sample size of 6, df = 5. You'll need to find the t value corresponding to α/2 = 0.005 (0.01/2) in a t-distribution table or use invT(0.995, 5) on a calculator.
• b) For a 99.9% confidence level with a sample size of 16, df = 15. You'll need to find the t value corresponding to α/2 = 0.0005 (0.001/2) in a t-distribution table or use invT(0.9995, 15) on a calculator.
• c) For a 99.8% confidence level with a sample size of 10, df = 9. You'll need to find the t value corresponding to α/2 = 0.001 (0.002/2) in a t-distribution table or use invT(0.999, 9) on a calculator.