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 sample confidence interval for ( 1 2 ) Start with the pivotal statistic and show that for large samples, Z is approximately a standard normal random variable Substitute s 2 1 for 1 2 and s 2 2 for 2 2 (why can you do this ) and follow the pivotal method of Example 7 6 (91 Y2) (H1 uz) Z 3 ( In) n2

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Question: 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

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

(91 - Y2) (H1 - uz) Z = + 3 ( In) 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.

(91 - Y2) (H1 - uz) Z = + 3 ( In) n2

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