Question

Find the critical values χ21−α/2 and χ2α/2 for a 90​% confidence level and a sample size of n=15.

Answer 1

To find the chi-square critical values for a 90% confidence level with a sample size of 15, we calculate the degrees of freedom (14) and then use a chi-square distribution table or calculator to find the critical values that correspond to the 5th and 95th percentiles for these degrees of freedom.

Step-by-step explanation:

To find the critical values for a 90% confidence level with a sample size of n=15, we first need to determine the degrees of freedom (df). Since the sample size is 15, the degrees of freedom for the chi-square distribution is df = n - 1 = 15 - 1 = 14.

For the 90% confidence level, the alpha (α) level is 100% - 90% = 10%, so we have α = 0.10. This alpha is split across the two tails of the distribution, so we have α/2 = 0.05 on each tail. We are interested in finding the critical values χ^2_{1-α/2} and χ^2_{α/2}.

To find these values, we use a chi-square distribution table, looking up the critical values corresponding to α/2 = 0.05 and 1-α/2 = 0.95 for df = 14. These are the values that cut off the top 5% and the bottom 5% of the distribution, respectively.

If you do not have a table, these values can also be found using statistical software or an online chi-square calculator.

Answer 2

To determine the critical values for a 90% confidence interval with a sample size of 15, calculate the degrees of freedom (n-1) and then use a chi-square distribution table to find the values at the 0.025 and 0.975 percentiles for 14 degrees of freedom.

Step-by-step explanation:

The question asks to find the critical values χ1−α/2 and χ2α/2 for a 90% confidence level and sample size of n=15. This relates to the concept of the chi-square (χ²) distribution in statistics, which is used to create confidence intervals around variance estimates in a population.

To find these critical values, one would first need to determine the degrees of freedom, which is the sample size minus 1 for a chi-square distribution (χ²), so we have df = 15 - 1 = 14. Then, since we are working with a 90% confidence interval, this means 5% is in the two tails (α/2 = 0.05/2 = 0.025 in each tail). Using a chi-square distribution table or an appropriate calculator/software, we would look up the values at the 0.025 and 0.975 percentiles for 14 degrees of freedom to find the critical values.