Find the critical values X2 1-a/2 and X2 a/2 for a 95% confidence level and a sample size of n=10.
X2 1-a/2 =
X2 a/2 =
Find the critical values X21-a/2and X2a/2 for a 95% confidence level and a sample size of n=25
To find the critical values X2 1-a/2 and X2 a/2 for a 95% confidence level and a sample size of n=10, we need to refer to the Chi-Square distribution table.
Step 1: Determine the degrees of freedom.
The degrees of freedom for a Chi-Square distribution depend on the sample size (n) and the number of parameters estimated in the population. For a sample size of n=10, and assuming we are estimating only the population variance (not the mean), the degrees of freedom can be calculated as (n-1) = (10-1) = 9.
Step 2: Determine the confidence level.
In this case, we are given a 95% confidence level.
Step 3: Identify the critical values.
The critical values for a Chi-Square distribution represent the boundary values that divide the distribution into areas corresponding to the desired confidence level.
Using the Chi-Square distribution table, we can find two critical values:
- X2 1-a/2 represents the upper critical value, which is associated with the upper (1-a/2) area (in this case, 1-0.05/2 = 0.975).
- X2 a/2 represents the lower critical value, which is associated with the lower (a/2) area (in this case, 0.05/2 = 0.025).
Referring to the Chi-Square distribution table with 9 degrees of freedom:
The value at the 0.975 (1-0.025) column and 9 degrees of freedom is approximately 21.67. This is the X2 1-a/2 critical value.
The value at the 0.025 column and 9 degrees of freedom is approximately 2.7. This is the X2 a/2 critical value.
Therefore, the critical values for a 95% confidence level and a sample size of n=10 are:
X2 1-a/2 = 21.67
X2 a/2 = 2.7