Differences of Random Variables Questions

[donkeybutt]

Hi guys, I need help with this question.

Suppose X has a normal distribution with mean u1 and known standard deviation 7.
Suppose Y has a normal distribution with mean u2 and known standard deviation 9.
Suppose we have a random sample of size 6 from the X distribution. The sample mean xbar is 24.
Suppose we have a random sample of size 8 from the Y distribution. The sample mean ybar is 27.

What is an appropriate value of zstar for a 95% confidence interval for u1 – u2?

Since the confidence interval has to be 95% is subtracted .95 from 1 to get .05 which is alpha. I then divided by 2. The z-score I got was 1.959964.

1-.95 = 0.05
0.5/2 = 0.025
z-score of 0.025 = 1.959964

Create a 95% confidence interval for u1 - u2.

24-27-1.959964*squareroot(49/6+81/8) and 24-27+1.959964*squareroot(49/6+81/8)

I did this and got (-11.382523 and 5.382523)

Calculate the variance of Xbar - Ybar

Now this third question is why I'm here. I have absolutely no clue how to solve this. Can someone offer me some help? Does anyone know the formula I can use to solve this and walk me through it? (Worried)(Poolparty)

 

[Valkarie]

The variance of Xbar - Ybar can be calculated using the following formula:Var(Xbar - Ybar) = Var(Xbar) + Var(Ybar) - 2*Cov(Xbar, Ybar)Where Var(Xbar) is the variance of the sample mean of X and Var(Ybar) is the variance of the sample mean of Y. Cov stands for covariance and is a measure of how two variables are related.To calculate the variance of Xbar, we use the formula Var(Xbar) = σ^2/n, where σ is the standard deviation of X (7 in this case) and n is the sample size (6 in this case). Therefore, Var(Xbar) = 49/6.Similarly, to calculate the variance of Ybar, we use the formula Var(Ybar) = σ^2/n, where σ is the standard deviation of Y (9 in this case) and n is the sample size (8 in this case). Therefore, Var(Ybar) = 81/8.Finally, to calculate the covariance between Xbar and Ybar, we use the formula Cov(Xbar, Ybar) = σxy/n, where σxy is the covariance between X and Y and n is the sample size. Since we don't know the covariance between X and Y, we can assume that it is 0 and therefore Cov(Xbar, Ybar) = 0.Putting everything together, we get:Var(Xbar - Ybar) = Var(Xbar) + Var(Ybar) - 2*Cov(Xbar, Ybar)= 49/6 + 81/8 - 0= 105/12Therefore, the variance of Xbar - Ybar is 105/12.