Let X₁ be a random variable from a distribution with a mean of 8 and a standard deviation of 6, and X₂ be a random variable from a distribution with a mean of 7 and a standard deviation of 5.

(i) For independent samples of sizes n₁ = 3 and n₂ = 6, find the mean and standard deviation of x bar₁ - x bar₂.

(ii) Do X₁ and X₂ have to be normally distributed for your answer to part (i) to be valid? Explain.

(iii) Can you conclude that the variable x bar₁ - x bar₂in part (i) is normally distributed? Explain.

(iv) Suppose X₁ and X₂ are normally distributed, determine the percentage of all pairs of independent samples of sizes n₁ = 3 and n₂ = 6 with the property that the difference between the sample means, x bar₁ - x bar₂, is between -9 and 11.