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

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

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 ~

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 ~

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 ~

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

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 ~

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 ~

Rejection regions f 120-2 +2.131-2.131

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

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

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 ~

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 

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 ~

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 ~

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

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 ~

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

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 ~

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

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

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 ~

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

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