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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X=25

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 F-distribution.

The F-statistic is given by F = (s1^2 / σ1^2) / (s2^2 / σ2^2), where s1^2 and s2^2 are the sample variances, and σ1^2 and σ2^2 are the population variances.

Given that the population variances are σ1^2 = 20 and σ2^2 = 36, we can calculate the F-statistic for the two samples.

F = (s1^2 / 20) / (s2^2 / 36)

Now, we need to find the probability that the variance of the first sample is more than twice the variance of the second sample, or P(F > 2).

To calculate this probability, we need the degrees of freedom for the two samples. The degrees of freedom for the first sample is given by (n1 - 1), and the degrees of freedom for the second sample is given by (n2 - 1), where n1 = 8 and n2 = 10.

Using a table or a statistical calculator with the degrees of freedom, we can find the probability P(F > 2) by looking up the value in the F-distribution table or using a calculator.

Alternatively, we can also use software like Excel to calculate the probability directly by using the F.DIST.RT function.

Note: The probability will depend on the specific values of the sample variances, and the sample sizes.

To find the probability that the variance of the first sample is more than twice the variance of the second sample, we can start by using the fact that the sample variance follows a chi-square distribution.

Let's assume that X₁ and X₂ are the random variables for the sample variances of the first and second samples, respectively.

Given that the sample sizes are 8 and 10, and the population variances are 20 and 36, we can calculate the degrees of freedom for each sample variance as follows:

For X₁, the degrees of freedom (df₁) would be (n₁ - 1) = (8 - 1) = 7.
For X₂, the degrees of freedom (df₂) would be (n₂ - 1) = (10 - 1) = 9.

According to the property of chi-square distribution, if Z is a chi-square random variable with v degrees of freedom, then the random variable Y = 2Z also follows a chi-square distribution with v degrees of freedom, but its mean and variance are 2v and 4v, respectively.

Now, we want to calculate the probability that the variance of the first sample (X₁) is more than twice the variance of the second sample (X₂). This can be expressed as P(X₁ > 2X₂).

To solve this, we need to transform the variables X₁ and X₂ into standard chi-square distributions.

Let Y₁ = X₁ / 20 (scaling factor for the first sample variance)
Let Y₂ = X₂ / 36 (scaling factor for the second sample variance)

Since Y₁ and Y₂ are scaled versions of X₁ and X₂, respectively, the probability we are interested in is now P(Y₁ > 2Y₂).

Next, we need to calculate the cumulative distribution function (CDF) values for Y₁ and 2Y₂ using the chi-square distribution tables or statistical software.

Finally, subtracting the CDF value of 2Y₂ from the CDF value of Y₁ will give us the desired probability:

P(X₁ > 2X₂) = P(Y₁ > 2Y₂) = P(Y₁) - P(2Y₂)

By using the chi-square distribution tables or statistical software, you can find the corresponding CDF values for Y₁ and 2Y₂, and subtract them to get the probability.