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Inferences About Means of Single Samples Chapter 10 Homework: 1-6

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Evaluating Hypotheses About Means n Evaluating hypothesis about population l taking a single sample l Does it likely come from population n Test statistics z test if known t test if unknown ~

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Steps in Hypothesis Evaluation 1. State null & alternative hypotheses 2. Set criterion for rejecting H 0 3. collect sample; compute sample statistic & test statistic 4. Interpret results l If reject H 0 u evaluate practical significance n Steps 1 & 2 before collecting data ~

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1. State null & alternative hypotheses unknown l calculate X, s, & s X from sample l use t test n Survey: college students study 21 hr/wk l Do Coe students study 21 hrs/week? l Select sample (n = 16) n Nondirectional hypothesis: H 0 : = 21; H 1 : 21 l reject H 0 if increase or decrease ~

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120-2 f What does distribution of sample statistic look like if H o true? If H o is false?

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2. Set Criterion for Rejecting H 0 n Determine critical value of test statistic l Directionality: l df = = l t CV = n Defines rejection region ~

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2. Set Criterion for Rejecting H 0 n Rejection region l area of distribution beyond critical value l for test statistic u Also sample statistic l Reject H 0 if t obs falls in rejection region ~

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Rejection regions f 120-2 +2.131-2.131

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3. Collect sample & compute statistics n Compute sample statistics u Mean u Standard deviation u Standard error of mean n Observed value of test statistic n General form test statistic = sample statistic - population parameter standard error of sample statistic

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3. Collect sample & compute statistics n Use sample statistics to compute test statistic l X = 24.63; s = 7.78, s X = 1.94 n Test statistic

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4. Interpret Results n Is t obs is beyond t CV ? l Is it in rejection region? l NO. 1.87 < 2.131 l then “accept” H 0 l Coe students study 21 hrs/wk n No significant difference l does not mean they are equal l not sufficient data to reject n Practical significance not an issue ~

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A Directional Hypothesis unknown: same question l evidence from prior surveys that Coe students study more than 21 hrs per week H 1 = experimental hypothesis can use directional hypotheses

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A Directional Hypothesis 1. State H 0 & H 1 H 0 : < 21 u Coe students study less than or equal to 21 hrs per week H 1 : > 21 u Coe students study more than 21 hrs per week ~

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A Directional Hypothesis 2. Set criterion for rejecting H 0 =.05, level of significance l directional (one-tailed) test l df = 15 l t CV = 1.753 ~

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A Directional Hypothesis 3. Collect sample & compute statistics l X = 24.63; s = 7.78, s X = 1.94 l test statistic = t obs

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A Directional Hypothesis 3. Interpret results l Is t obs in rejection region? l t obs > t CV 1.87 > 1.753 n Reject H 0 l accept H 1 l Coe students study more than 21 hours per week ~

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Practical Significance n Statistical significance? l YES n Practical significance? l MAYBE ~

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Practical Significance: Effect size n Magnitude of the result (difference) n Raw effect size l measured on scale of original data X obs - = 24.63 - 21 = 3.63 l Coe students study 3.63 hours per week longer than the national average ~

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Practical Significance: Effect size n Effect size index l compare effect size for variables using different scales (e.g. GRE, ACT) l divide difference by s nondirectionaldirectional

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=.47 standard deviations above the mean 120-2 f d standard deviations Effect size index

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Practical Significance: Effect size n Is effect size practically significant? l.5 considered moderate effect size e.g., Is it worth using a new statistics textbook that test scores d =.5? l Ultimately we must make decision ~

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When Is Known n Usually not the situation l calculate X from sample l use z test l degrees of freedom not relevant l find z CV in z table

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Practical Significance: Effect size Effect size index: is known nondirectionaldirectional

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