7.1 Comparing Two Population Means: Independent Sampling

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Derek Ramsey
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1 University of California, Davis Department of Statistics Summer Session II Statistics 13 September 4, 01 Lecture 7: Comparing Population Means Date of latest update: August Comparing Two Population Means: Iepeent Sampling Properties of the Sampling Distribution of ( x 1 x 1. The mean of the sampling distribution of ( x 1 x is (µ 1 µ.. If the two samples are iepeent, the staard deviation of the sampling distribution is σ (x1 x = ( σ 1 + σ where σ 1 a σ are the variances of the two populations being sampled a a are the respectively sample sizes. We also refer to σ (x1 x as the staard error of the statistic ( x 1 x. 3. By the central limit theorem, the sampling distribution of ( x 1 x is approximately normal for large samples Large Sample Confidence Interval for (µ 1 µ σ1 ( x 1 x ± z α/ σ ( x1 x = ( x 1 x ± z α/ + σ s 1 ( x 1 x ± z α/ + s Large-Sample Test of Hypothesis for (µ 1 µ One-Tailed Test Two-Tailed Test H 0 : (µ 1 µ = D 0 H 0 : (µ 1 µ = D 0 H a : (µ 1 µ < D 0 H a : (µ 1 µ D 0 [or H a : (µ 1 µ > D 0 ] where D 0 = Hypothesized difference between the means (this difference is often hypothesized to be equal to 0 1

2 Test statistic: z = ( x 1 x D 0 σ ( x1 x where σ ( x1 x = σ 1 + σ s 1 + s Rejection region: z < z α Rejection region: z > z α/ or z > z α when H a : (µ 1 µ > D 0 Coitions Required for Valid Large-Sample Inferences about (µ 1 µ 1. The two samples are raomly selected in an iepeent manner from the two target populations.. The sample sizes, a, are both large (i.e., 30 a 30. (By the central limit theorem, this coition guarantees that the sampling distribution of ( x 1 x will be approximately normal, regardless of the shapes of the uerlying probability distributions of the populations. Also, σ 1 a σ will provide good approximations to σ 1 a σ when both samples are large Small-Sample Confidence Interval for (µ 1 µ : Iepeent Samples ( 1 ( x 1 x ± t α/ s p + 1 where s p = ( 1s 1 +( 1s ( + a t α/ is based on ( + degrees of freedom. Small-Sample Test of Hypothesis for (µ 1 µ : Iepeent Samples One-Tailed Test Two-Tailed Test H 0 : (µ 1 µ = D 0 H 0 : (µ 1 µ = D 0 H a : (µ 1 µ < D 0 H a : (µ 1 µ D 0 [or H a : (µ 1 µ > D 0 ] Test statistic: t = ( x 1 x D 0 s p ( Rejection region: t < t α Rejection region: t > t α/ or t > t α when H a : (µ 1 µ > D 0 where t α a t α/ are based on ( + degrees of freedom. Coitions Required for Valid Small-Sample Inferences about (µ 1 µ 1. The two samples are raomly selected in an iepeent manner from the two target populations.. Both sampled populations have distributions that are approximately normal. 3. The population variances are equal (i.e., σ 1 = σ.

3 7.1.3 Approximate Small-Sample Procedures when σ 1 σ Equal Sample Sizes ( = = n Confidence interval: ( x 1 x ± t α/ (s 1 + s /n Test statistic for H 0 : (µ 1 µ = 0 : t = ( x 1 x / (s 1 + s /n where t is based on v = + = (n 1 degrees of freedom. Unequal Sample Sizes ( Confidence interval: ( x 1 x ± t α/ (s 1 / + (s / Test statistic for H 0 : (µ 1 µ = 0 : t = ( x 1 x / (s 1/ + (s / where t is based on degrees of freedom equal to v = (s 1/ + s / (s 1 / + (s / 1 1 Note: The value of v will generally not be an integer. integer to use the t-table. Rou v down to the nearest 7. Comparing Two Population Means: Paired Difference Experiments Paired Difference Confidence Interval for µ d = µ 1 µ Large Sample σ d x d ± z α/ x d ± z α/ s d Small Sample x d ± t α/ s d where t α/ is based on (n d 1 degrees of freedom. 3

4 Paired Difference Test of Hypothesis for µ d = µ 1 µ Large Sample Test statistic: z = x d D 0 σ d / n d x d D 0 s d / n d Small Sample Test statistic: t = x d D 0 s d / n d One-Tailed Test Two-Tailed Test H 0 : µ d = D 0 H 0 : µ d = D 0 H a : µ d < D 0 H a : µ d D 0 [or H a : µ d > D 0 ] Rejection region: z < z α Rejection region: z > z α/ or z > z α when H a : µ d > D 0 Rejection region: t < t α Rejection region: t > t α/ or t > t α when H a : µ d > D 0 where t α a t α/ are based on (n d 1 degrees of freedom Coitions Required for Valid Large-Sample Inferences about µ d 1. A raom sample of differences is selected from the target population of differences.. The sample size n d is large (i.e., n d 30. (By the central limit theorem, this coition guarantees that the test statistic will be approximately normal, regardless of the shape of the uerlying probability distribution of the population. Coitions Required for Valid Small-Sample Inferences about µ d 1. A raom sample of differences is selected from the target population of differences.. The population of differences has a distribution that is approximately normal. 4

5 7.3 Determining the Sample Size Determination of Sample Size for Comparing Two Means Iepeent Raom Samples To estimate (µ 1 µ to within a given sampling error SE a with confidence level (1 α, use the following formula to solve for equal sample sizes that will achieve the desired reliability: = = (z α/ (σ 1 + σ (SE You will need to substitute estimates for the values of σ 1 a σ before solving for the sample size. These estimates might be sample variances s 1 a s from prior sampling (e.g., a pilot study, or from an educated (a conservatively large guess based on the range that is, s R/4. Paired Difference Experiment To estimate µ d to within a given sampling error SE a with confidence level (1 α, use the following formula to solve for n d that will achieve the desired reliability: n d = (z α/ σ d (SE You will need to substitute an estimate of σd before solving for the sample size. This estimate might be the sample variance s d from prior sampling (e.g., a pilot study, or from an educated (a conservatively large guess based on the range that is, s d R/4. 5

Two Populations Hypothesis Testing

Two Populations Hypothesis Testing Two Proportions (Large Independent Samples) Two samples are said to be independent if the data from the first sample is not connected to the data from the second sample.