Question: 1. Let X 1 be a random variable from a distribution with a mean of 8 and a standard deviation of 6 and X 2

1.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.

(a) For independent samples of sizes n₁ = 3 and n₂ = 6, find the mean and standard deviation of . (3 marks)

(b) Do X₁ and X₂ have to be normally distributed for your answer to part (a) to be valid? Explain. (2 marks)

(c) Can you conclude that the variable in part (a) is normally distributed? Explain. (3 marks)

(d) 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, , is between -9 and 11. (5 marks)

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