Suppose you have a sample of 6 observations from a normal population. The sample
variance is equal to 4. Find a 90% lower confidence interval for the variance of the
population.
To find the lower confidence interval for the variance of the population, we can use the Chi-square distribution.
The formula to calculate the confidence interval for the variance is:
( (n-1)*s^2 ) / X^2 < variance < ( (n-1)*s^2 ) / X^2
where:
- n is the sample size
- s^2 is the sample variance
- X^2 is the chi-square value
Given:
- Sample size (n) = 6
- Sample variance (s^2) = 4
- Confidence level = 90%
- Degrees of freedom = n-1 = 5
Step 1: Determine the chi-square values for the lower and upper confidence levels.
Since we want a 90% confidence interval, we need to find the chi-square values that correspond to a significance level of (1 - confidence level) / 2 on each tail. In this case, it would be (1 - 0.90) / 2 = 0.05/2 = 0.025 for each tail.
Using a chi-square distribution table or a calculator, the chi-square value corresponding to a significance level of 0.025 with 5 degrees of freedom is approximately 11.070.
Step 2: Plug in the values into the formula.
Lower confidence interval = ( (n-1)*s^2 ) / X^2
= ( (6-1)*4 ) / 11.070
= (5*4) / 11.070
= 20 / 11.070
= 1.8055
Therefore, the 90% lower confidence interval for the variance of the population is 1.8055.
To calculate a confidence interval for the variance of a population, we can use the chi-squared distribution.
Here's how you can do it step by step:
1. Determine the degrees of freedom (df) for the chi-squared distribution. For variance, the degrees of freedom is equal to n-1, where n is the number of observations in the sample. In this case, we have 6 observations, so df = 6-1 = 5.
2. Determine the critical values for the chi-squared distribution corresponding to a 90% confidence level and the degrees of freedom. You can use a chi-squared table or a statistical software to find the critical values. For a 90% confidence level with df = 5, the critical values are approximately 2.571 for the lower tail.
3. Calculate the chi-squared statistic, which is equal to (n-1) * sample variance divided by the population variance. In this case, the sample variance is given as 4. So the chi-squared statistic can be calculated as (6-1) * 4 / population variance.
4. Set up the confidence interval formula using the chi-squared statistic, critical value, and degrees of freedom. The formula for a lower confidence interval is: (n-1) * sample variance / chi-squared critical value.
Plugging in the values, we have (5*4) / 2.571 = 7.76.
5. Convert the chi-squared statistic to a confidence interval for the population variance. Since we are looking for a lower confidence interval, we only need to find the lower bound. In this case, the lower bound is the chi-squared statistic itself, which is 7.76.
So the 90% lower confidence interval for the variance of the population is [7.76, ∞)