9-2 Two Means: Independent Samples 461 The requirements for this case are the same as in Part 1, except the first requirement is that s1 and s2 are not known but they are assumed to be equal. Confidence intervals are found by evaluating 1x1 - x22 - E 6 m1 - m2 6 1x1 - x22 + E with the following margin of error E. Margin of Error for Confidence Interval E = ta>2Bs2 p n1 + s2 p n2 where s2 p is as given in the test statistic above, and df = n1 + n2 - 2. When Should We Assume That S1 = S2? If we use randomness to assign subjects to treatment and placebo groups, we know that the samples are drawn from the same population. So if we conduct a hypothesis test assuming that two population means are equal, it is reasonable to assume that the samples are from populations with the same standard deviations (but we should still check that assumption). Advantage of Pooling Sample Variances The advantage of this alternative method of pooling sample variances is that the number of degrees of freedom is a little higher, so hypothesis tests have more power and confidence intervals are a little narrower. Alternative Method Used When S1 and S2 Are Known In reality, the population standard deviations s1 and s2 are almost never known, but if they are somehow known, the test statistic and confidence interval are based on the normal distribution instead of the t distribution. The requirements are the same as those given in Part 1, except for the first requirement that s1 and s2 are known. Critical values and P-values are found using technology or Table A-2, and the test statistic for this case is as follows: Test Statistic z = 1x1 - x22 - 1m1 - m22 Bs 2 1 n1 + s 2 2 n2 Confidence intervals are found by evaluating 1x1 - x22 - E 6 m1 - m2 6 1x1 - x22 + E, where: Margin of Error for Confidence Interval E = za>2Bs 2 1 n1 + s 2 2 n2 What if One Standard Deviation Is Known and the Other Is Unknown? If s1 is known but s2 is unknown, use the procedures in Part 1 of this section with these changes: Replace s1 with the known value of s1 and use the number of degrees of freedom found from the expression below. (See “The Two-Sample t Test with One Variance Unknown,” by Maity and Sherman, The American Statistician, Vol. 60, No. 2.) df = a s 2 1 n1 + s2 2 n2b 2 1s2 2>n22 2 n2 - 1 Recommended Strategy for Two Independent Means Here is the recommended strategy for the methods of this section: Assume that S1 and S2 are unknown, do not assume that S1 = S2, and use the test statistic and confidence interval given in Part 1 of this section. Using Statistics to Identify Thieves Methods of statistics can be used to determine that an employee is stealing, and they can also be used to estimate the amount stolen. For comparable time periods, samples of sales have means that are significantly different. The mean sale amount decreases significantly. There is a significant increase in “no sale” register openings. There is a significant decrease in the ratio of cash receipts to checks. (See “How to Catch a Thief,” by Manly and Thomson, Chance, Vol. 11, No. 4.) t th t
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