Let y 1 and s 1 2 be the sample mean and sample variance, respectively, of n 1 observations randomly selected from a population with mean 1 and variance 1 2 Similarly, define y 2 and s 2 2 for an independent random sample of n 2 observations from a population with mean 2 and 2 2 Derive a large samp...

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Let y 1 and s 1 2 be the sample mean and sample variance, respectively, of n

Question:

Let y̅₁ and s₁² be the sample mean and sample variance, respectively, of n₁ observations randomly selected from a population with mean μ₁ and variance σ₁². Similarly, define y̅₂ and s₂² for an independent random sample of n₂ observations from a population with mean μ₂ and σ₂². Derive a large-sample confidence interval for (μ₁ - μ₂). Start with the pivotal statistic

and show that for large samples, Z is approximately a standard normal random variable. Substitute s²₁ for σ₁² and s₂² for σ₂² (why can you do this?) and follow the pivotal method of Example 7.6.