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Mean difference between independent samples, sampling distribution, hypothesis testing, test statistics and p-values

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STAT 301
Brett Hunter

17 October Mean difference between independent samples Goal is to compare the mean of the variable for one population to the mean of the variable in the second So our parameter of interest is μ – μ1 2 We can have two samples of different sizes, n and n .1 2 Assumption Both n a1d n are2greater than or equal to 30 (or both populations normal/combo) Sample 1 and Sample 2 selected independently of each other Sampling Distribution We want to make inferences about μ – μ 1 2 Take 2 independent samples Find x 1 , the point estimate for μ 1 Find x 2 , the point estimate for μ 2 x 1 - x2 is the point estimate for μ - μ 1 2 We need the sampling distribution of x 1 - x2 in order to construct hypothesis tests and confidence intervals If the random samples are independent of one another and n and n are b1th larg2 (≥ 2 2 x x σ1 σ2 30), then 1 - 2 ~ N(μ –1μ , 2 n + n ) 1 2 Hypothesis Testing Most common hypotheses compare the difference to 0, but could also compare to some other number. Hypotheses Upper tail (is μ 1arger than μ ?)2 H0: μ1≤ μ 2 HA: μ1> μ 2 OR H : μ – μ ≤ 0 0 1 2 HA: μ1- μ 2 0 Lower tail (is μ 1maller than μ ?) 2 H0: μ1≥ μ 2 HA: μ1< μ 2 OR H0: μ1– μ ≥20 HA: μ1– μ <20 Two tail (are μ 1nd μ dif2erent?) H0: μ1= μ 2
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