# Example: We wish to compare the cube compressive strength...

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Sets 28 and 29 Instead of drawing samples from one population, we may take random samples from two populations so that we can carry out some sort of comparison. We may wish to compare μ 1 and μ 2 , the population means for populations 1 and 2 . We do so by examining the difference, μ 1 - μ 2 . Example: Suppose we wish to compare the μ 1 , the mean lead content (in ppm ) per mL in Victoria tap water with μ 2 , the mean lead content (in ppm ) per mL in Vancouver tap water. If the means are equal, then μ 1 - μ 2 = 0 . If the means are different, then μ 1 - μ 2 6 = 0 . If the lead content is higher in Victoria, then μ 1 - μ 2 > 0 . If the lead content is higher in Vancouver, then μ 1 - μ 2 < 0 . If the lead content is higher in Victoria by at least 4 ppm than in Vancouver, then μ 1 - μ 2 > 4 If the lead content is higher in Vancouver by at least 2 ppm than in Victoria, then μ 1 - μ 2 < - 2 1
The point estimate for μ 1 - μ 2 we will use is x 1 - x 2 . The following pivotal quantities apply to a variety of cases where two samples are drawn independently from two populations. As before, any confidence interval we construct will have the form estimate ± (c.v.)(e.s.e.) All of the pivotal quantities below have the form estimate - parameter e . s . e . . Large Sample Size Procedures: Z = ( x 1 - x 2 ) - ( μ 1 - μ 2 ) s s 2 1 n 1 + s 2 2 n 2 Assumptions: Independent random samples from two populations. Both sample sizes are large ( n 1 40 , n 2 40) and the population standard deviations are unknown. Populations may have any distribution. 2
Example: We wish to compare the cube compressive strength (in N/mm 2 ) of two types of concrete. The summary statistics are as follows: sample size sample mean sample sd Type A 70 31.9 1.4 Type B 50 35.6 2.1 Test the research hypothesis that the two types of concrete have different mean cube compressive strengths. 3
Small Sample Size Procedures: In the case where n 1 or n 2 (or both) are small, there are two choices of test statistic. This choice depends on whether or not we assume that σ 1 = σ 2 . While there are hypothesis tests that we could carry out to investigate whether or not σ 1 = σ 2 , we will use the following rule for our class: Using s 1 , s 2 , calculate the larger standard deviation divided by the smaller If this value is less than 1.4 , we assume σ 1 = σ 2 . If it is greater than 1.4 , we assume σ 1 6 = σ 2 . Note that we only need to decide whether or not σ 1 = σ 2 when the sample size is not large. 4
Pooled Procedures: t n 1 + n 2 - 2 = ( x 1 - x 2 ) - ( μ 1 - μ 2 ) s ( n 1 - 1) s 2 1 + ( n 2 - 1) s 2 2 n 1 + n 2 - 2 1 n 1 + 1 n 2 Assumptions: Independent random samples from two populations. At least one of the sample sizes is small, and the population standard deviations are unknown. We know that (or assume that) σ 1 = σ 2 Both populations have normal (or approximately normal) distribu- tion. Comments: The value ( n 1 - 1) s 2 1 + ( n 2 - 1) s 2 2 n 1 + n 2 - 2 is sometimes denoted by s 2 p , and is called the pooled variance estimate . Recall that for pooled procedures, we assumed that σ 1 = σ 2 . The value of s 2 p is the estimate for both σ 2 1 and σ 2 2 . 5
Unpooled Procedures: t γ = ( x 1 - x 2 ) - ( μ 1 - μ 2 ) s s 2 1 n 1 + s 2 2 n 2 where γ