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# 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 simplest research situation l taking a single sample n Test statistics z test if  known t test if  unknown n If reject H 0 l evaluate practical significance ~

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 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 Average college students study 21 hours per week? l Do Coe students study 21 hrs/week? n Nondirectional hypothesis: n = 16 H 0 :  = 21; H 1 :   21 l reject H 0 if increase or decrease ~

1. State null & alternative hypotheses n 3 important distributions variable: X l sample statistic: X central limit theorem l test statistic: z or t known probabilities n Distributions show all possible values of variable l assuming H 0 is true

0 +1 +2 -2 What does distribution of sample statistic look like if H o true? If H o is false?

1. State null & alternative hypotheses n Test statistic n General form test statistic = sample statistic - population parameter standard error of sample statistic

2. Set Criterion for Rejecting H 0 n Directionality & level of significance n X obs = computed sample statistic l same as randomly drawing a single X from sampling distribution of means n X CV = critical value of the statistic l set in advance l beginning of the rejection region area in tails of distribution l if X obs lies beyond, reject H 0 ~

2. Set Criterion for Rejecting H 0 n Computing critical values of statistic X CV =  + t CV (s  X ) l same as confidence intervals l X CV = upper & lower limits l reject H 0 if beyond n *Critical value of test statistic l df = 15 l t.05 = 2.131 (nondirectional) ~

2. Set Criterion for Rejecting H 0 n Rejection region l portion of distribution beyond critical value area in tails l for sample statistic or test statistic n Level of significance if  =.05 l nondirectional:.025 in each tail.025 +.025 =.05 ~

Rejection regions f 120-2 +2.131-2.131

2. Set Criterion for Rejecting H 0 n Test statistic l observed value computed from sample l critical value criterion set in advance depends on  (level of significance) & directionality nondirectional: t.05 = 1.96 if directional: t.05 = 1.645 ~

2. Set Criterion for Rejecting H 0 n Decision l if t obs is beyond t CV,then reject H 0 l if not, “accept” H 0

3. Collect sample & compute statistics n Collect data & 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 NO. 1.87 < 2.131 l then “accept” H 0 l Students study about 21 hrs per week. 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 experimental hypothesis = H 1 can use directional hypotheses 

A Directional Hypothesis 1. State H 0 & H 1 H 0 :  < 21 Coe students study less than or equal to 21 hrs per week H 1 :  > 21 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 critical value for area =.05 (one-tailed) ~

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 t obs > t CV 1.87 > 1.753 n Reject H 0 l accept H 1 Coe students study more than 21 hours per week ~

Practical Significance n Statistical significance? l YES n Practical significance? l MAYBE n Determining practical significance l effect size ~

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

0 +1 +2 -2 d =.47 standard deviations above the mean

Practical Significance: Effect size n Is effect magnitude 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 l using our expertise l considering many factors ~

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 use  X

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

Relationship to Confidence Intervals n Nondirectional tests equivalent to CI Level of significance:  =.05 Level of confidence: 1-  =.95 95% confident that true value of  falls within interval l if it does: H 0 is true n If falls outside CI: reject H 0 ~

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