Two simple of sizes 8&10 are drawn from two normally distributed population having variance 20 and 36 respectively. Find the probality that the variance of the first sample is more than twice the variance of the second sample
To find the probability that the variance of the first sample is more than twice the variance of the second sample, we need to make use of the Chi-Square distribution.
Given:
- Sample size of the first population (n1) = 8
- Sample size of the second population (n2) = 10
- Variance of the first population (var1) = 20
- Variance of the second population (var2) = 36
Step 1: Calculate the Chi-Square statistic for each sample.
- Chi-Square statistic for the first sample (chi1): (n1 - 1) * var1 / σ1^2
- Chi-Square statistic for the second sample (chi2): (n2 - 1) * var2 / σ2^2
(where σ1^2 and σ2^2 are the true variances of the populations)
Step 2: Calculate the degrees of freedom for each sample.
- Degrees of freedom for the first sample: df1 = n1 - 1 = 8 - 1 = 7
- Degrees of freedom for the second sample: df2 = n2 - 1 = 10 - 1 = 9
Step 3: Calculate the cumulative probability using the Chi-Square distribution.
- Probability that the variance of the first sample is more than twice the variance of the second sample: 1 - P(chi1 ≤ 2 * chi2)
To calculate this probability, you can consult a Chi-Square distribution table, or use statistical software or online calculators. These tools provide the cumulative probability associated with different Chi-Square values for given degrees of freedom.
Note: The calculations assume that the sample variances are unbiased estimators of the population variances, and that the populations are normally distributed.