Q- Two samples of sizes 8 and 10 are drawn from two normally distributed population having variance 20 and 36 respectively. Find the probability that the variance of the first sample is more than twice the variance of the second sample.
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To find the probability that the variance of the first sample is more than twice the variance of the second sample, we need to use the F-distribution.
The F-distribution is used to compare the variances of two independent normal populations. In this case, we have two samples from two normally distributed populations.
The F-statistic is defined as the ratio of the variances between the two samples. If the F-statistic is greater than 1, it means that the variance of the first sample is larger than the variance of the second sample.
To calculate the probability, we need to find the critical value of the F-statistic for a given level of significance (alpha), degrees of freedom for the numerator (v1), and degrees of freedom for the denominator (v2).
In this case, we have two samples of sizes 8 and 10, with variances of 20 and 36 respectively. The degrees of freedom for the numerator (v1) are equal to the sample size of the first sample minus 1 (8 - 1 = 7), and the degrees of freedom for the denominator (v2) are equal to the sample size of the second sample minus 1 (10 - 1 = 9).
Assuming a significance level (alpha) of 0.05, we need to find the critical value of the F-statistic with degrees of freedom v1 = 7 and v2 = 9.
Using statistical software or F-distribution tables, we can find that the critical value is approximately 2.50.
Now, we can calculate the probability using the cumulative distribution function (CDF) of the F-distribution. The probability can be calculated as P(F > 2.50), where F is the F-statistic.
Finally, we can use the cumulative distribution function of the F-distribution with the degrees of freedom v1 = 7 and v2 = 9, to find the probability that the variance of the first sample is more than twice the variance of the second sample.
Note: The exact calculation may vary depending on the software or tools you are using to perform the calculations. It is recommended to use statistical software or consult a statistician for precise calculations.