Let 𝑋1,…,𝑋5 and 𝑌1,…,𝑌8 be two independent simple random samples such as 𝑋𝑖~𝑁(𝑚 = 7,𝜎^2 = 50) and 𝑌𝑖~𝑁(𝑚 = 5,𝜎^2 = 24). Find a real number c such as 𝑃(𝑋̅−𝑌̅ > 𝑐) = 0.3

Should be an easy problem but I can’t understand if I’m solving it right. How to find this number c, that probability of 𝑋̅−𝑌̅ higher than c is equal 0.3? Please help

Solution: Here I found variance of means and their difference: Var(𝑋̅) = δ^2/n = 50/5 = 10;
Var(𝑌̅) = δ^2/n = 24/8 = 3; 𝑋̅ ~ N(7, 10);
𝑌̅ ~ N(5, 3); 𝑋̅−𝑌̅ ~ N(2, 13); (𝑋̅−𝑌̅-2)/(√13) ~ N(0, 1)

Let $X_1,\ldots,X_5$ and $Y_1,\ldots,Y_8$ be two independent simple random samples such that $X_i\sim\mathcal N(m = 7,\sigma^2 = 50)$ and $Y_i\sim\mathcal N(m = 5,\sigma^2 = 24)$. Find a real number $c$ such that $𝑃(\bar X−\bar Y > c) = 0.3$

Should be an easy problem but I can’t understand if I’m solving it right. How to find this number c, that probability of $\bar X−\bar Y$ higher than c is equal 0.3? Please help

My attempt: Here I found variance of means and their difference: $Var(\bar X) = \delta^2/n = 50/5 = 10$;
$Var(\bar Y) = \delta^2/n = 24/8 = 3$; $\bar X \sim \mathcal N(7, 10)$;
$\bar Y \sim \mathcal N(5, 3)$; $\bar X−\bar Y \sim \mathcal N(2, 13)$; $(\bar X−\bar Y -2)/(√13) \sim \mathcal N(0, 1)$

Let 𝑋1,…,𝑋5 and 𝑌1,…,𝑌8 be two independent simple random samples such as 𝑋𝑖~𝑁(𝑚 = 7,𝜎^2 = 50) and 𝑌𝑖~𝑁(𝑚 = 5,𝜎^2 = 24). Find a real number c such as 𝑃(𝑋̅−𝑌̅ > 𝑐) = 0.3

Should be an easy problem but I can’t understand if I’m solving it right. How to find this number c, that probability of 𝑋̅−𝑌̅ higher than c is equal 0.3? Please help

Solution: Var(𝑋̅) = δ^2/n = 50/5 = 10;
Var(𝑌̅) = δ^2/n = 24/8 = 3; 𝑋̅ ~ N(7, 10);
𝑌̅ ~ N(5, 3); 𝑋̅−𝑌̅ ~ N(2, 13); (𝑋̅−𝑌̅-2)/(√13) ~ N(0, 1)

Let 𝑋1,…,𝑋5 and 𝑌1,…,𝑌8 be two independent simple random samples such as 𝑋𝑖~𝑁(𝑚 = 7,𝜎^2 = 50) and 𝑌𝑖~𝑁(𝑚 = 5,𝜎^2 = 24). Find a real number c such as 𝑃(𝑋̅−𝑌̅ > 𝑐) = 0.3

Should be an easy problem but I can’t understand if I’m solving it right. How to find this number c, that probability of 𝑋̅−𝑌̅ higher than c is equal 0.3? Please help

Solution: Here I found variance of means and their difference: Var(𝑋̅) = δ^2/n = 50/5 = 10;
Var(𝑌̅) = δ^2/n = 24/8 = 3; 𝑋̅ ~ N(7, 10);
𝑌̅ ~ N(5, 3); 𝑋̅−𝑌̅ ~ N(2, 13); (𝑋̅−𝑌̅-2)/(√13) ~ N(0, 1)