Two samples of sizes 8 and 10 are drawn from two normally distributed
populations having variances 21.8 and 36, respectively. Find the probability that
the variance of the first sample is more than twice the variance of the second
sample. Interpret its value.
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 used to test if two samples have significantly different variances.
Let's define:
- n1 = number of observations in the first sample = 8
- n2 = number of observations in the second sample = 10
- v1 = variance of the first population = 21.8
- v2 = variance of the second population = 36
Now, we can calculate the F-ratio, which is the ratio of the variances of the two samples:
F = (v1 / n1) / (v2 / n2)
To find the probability that the variance of the first sample is more than twice the variance of the second sample, we want to find P(F > 2).
To calculate this probability, we need the cumulative distribution function (CDF) of the F-distribution. However, it's important to note that the CDF of the F-distribution depends on the degrees of freedom.
In this case, the degrees of freedom for the first sample is (n1 - 1) = 7, and the degrees of freedom for the second sample is (n2 - 1) = 9.
Using statistical software or a F-distribution table, we can find the probability P(F > 2) based on the degrees of freedom.
Interpreting the value of the probability will depend on the context of the problem and the specific hypothesis being tested. In general, a smaller probability indicates stronger evidence that the variance of the first sample is significantly larger than twice the variance of the second sample. On the other hand, a larger probability suggests weaker evidence for the difference in variances.