two sample of size 8 and 10 are drawn from two normally distributed populations having variances 20 and 36, respectively. find the probability 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 can use the F-distribution. The F-distribution is a probability distribution that is used to compare the variances of two populations.
The F-statistic is given by the formula:
F = (s1^2 / σ1^2) / (s2^2 / σ2^2)
Where:
- s1^2 is the sample variance of the first sample
- σ1^2 is the population variance of the first population
- s2^2 is the sample variance of the second sample
- σ2^2 is the population variance of the second population
In this case, the variances of the two populations are given as 20 and 36, respectively.
Let's assume that the null hypothesis is that the variance of the first sample is twice the variance of the second sample, i.e., σ1^2 = 2σ2^2.
Now, we can calculate the F-statistic using the given information:
F = (s1^2 / σ1^2) / (s2^2 / σ2^2)
= (s1^2 / 2σ2^2) / (s2^2 / σ2^2)
= (s1^2 / 2) / s2^2
To find the probability that the variance of the first sample is more than twice the variance of the second sample, we need to calculate the area under the F-distribution curve to the right of this F-statistic value.
Using a calculator or statistical software, you can input the degrees of freedom for the two samples (df1 = n1 - 1 and df2 = n2 - 1) along with the F-statistic value to find the probability. In this case, the first sample size is 8, so df1 = 8 - 1 = 7, and the second sample size is 10, so df2 = 10 - 1 = 9.
By plugging the values into an F-distribution calculator or using statistical software, you can find the probability that the variance of the first sample is more than twice the variance of the second sample.