2 Objectives Conduct one sample mean test Using Z statistics Using t statistics
3 Inferential Statistics Usage Researchers use inferential statistics to address two broad goals:Estimate the value of population parametersHypothesis testing
4 Distribution of Coin Tosses If you see 10 heads in a row,is it a fair coin?
5 Sample & PopulationThink of any sequence of throws as a sample from all possible throwsThink of all possible throws as the entire population.One-Sample Inferential Tests estimate the probability that a sample is representative of the total population (within +/- ~2 standard deviations of the mean, or the middle 95% of the distribution).
6 Logic of Hypothesis Testing Is the value observed consistent with the expected distribution?On average, 100 coin tosses should lead to 50/50 chance of heads.Some coin tosses will be outliers, giving significantly different results.Are differences significant or merely random variations?Statistics is the art of making sense of distributions
7 Logic of Hypothesis Testing The further the observed value is from the mean of the expected distribution, the more significant the difference
11 Probability of Membership in a Distribution Depends on locationMeanVarianceIt is a chance event
12 One-Sample TestsWe set a standard beyond which results would be rare (outside the expected sampling error)We observe a sample and infer information about the populationIf the observation is outside the standard, we reject the hypothesis that the sample is representative of the population
13 Random SamplingA simple random sampling procedure is one in which every possible sample of n objects is equally likely to be chosen.The principle of randomness in the selection of the sample members provides some protection against the sample unrepresentative of the population.If the population were repeatedly sampled in this fashion, no particular subgroup would be over represented in the sample.
14 Sampling Distribution The concept of a sampling distribution, allows us to determine the probability that the particular sample obtained will be unrepresentative.On the basis of sample information, we can make inference about the parent population.
15 Sampling Distribution Sampling Error.No sample will have the exact same mean and standard deviation as the populationSampling distribution of the meanIn research sampling error is often unknown since we do not have the population parametersA distribution of means of several different samples of our populationLess widely distributed than the populationUsually Normal
16 Population of IQ scores, 10-year olds µ=100σ=16n = 64Sample 1Sample 2Sample 3EtcIs sample 2 a likelyrepresentationof our population?
17 Distribution of Sample Means The mean of a sampling distribution is identical to mean of raw scores in the population (µ)If the population is Normal, the distribution of sample means is also NormalIf the population is not Normal, the distribution of sample means approaches Normal distribution as the size of sample on which it is based gets largerCentralLimitTheorem
18 Standard Error of the Mean The standard deviation of means in a sampling distribution is known as the standard error of the mean.It can be calculated from the standard deviation of observationsThe larger our sample size, the smaller our standard error
19 Statistical inference Sample ofobservationsEntire population ofobservationsRandom selectionParameterµ=?StatisticStatistical inference
20 Estimation Procedures Point estimatesFor example mean of a sample of 25 patientsNo information regarding probability of accuracyInterval estimatesEstimate a range of values that is likelyConfidence interval between two limit valuesThe degree of confidence depends on the probability of including the population mean
21 When Sample size is small … A constant fromStudent t Distributionthat depends on confidenceinterval and sample size
22 HYPOTHESIS TESTINGHygiene procedures are effective in preventing cold.State 2 hypotheses:Null: H0 : Hand-washing has no effect on bacteria counts.Alternative: Ha : Hand-washing reduces bacteria.The null hypothesis is assumed true: i.e., the defendant is assumed to be innocent.
23 TWO TYPES OF ERROR True False Reject H0 error correct decision Fail to Reject H0
24 Alpha & Beta Errors Decision Ho is True Ho is False Reject H0 α 1-β Fail to Reject H01-αβ
25 Two Types of Error in Admission to ICU Correct decisionsPatients admitted to ICU who would have failed if otherwisePatients denied admission who do fine in step down unitErrorsPatient admitted who does not need to be therePatient denied admission who needs to be there
26 Two Types of Error Alpha: α Beta: β Probability of Type I Error P (Rejecting Ho when Ho is true)Beta: βProbability of Type II ErrorP (Failing to reject Ho when Ho is false)
27 Power & Confidence Level 1- βProbability of rejecting Ho when Ho is falseConfidence level1- αProbability of failing to reject Ho when Ho is true
28 Steps in Test of Hypothesis Determine the appropriate testEstablish the level of significance:αDetermine whether to use a one tail or two tail testCalculate the test statisticDetermine the degree of freedomCompare computed test statistic against a tabled value
29 1. Determine Appropriate Test Level of measurementNumber of groups being comparedSample sizeExtent to which assumption for parametric tests have been metRelatively Normal distributionApproximately interval level variable
30 2. Establish Level of Significance α is a predetermined valueThe conventionα = .05α = .01
31 3. Determine Whether to Use One or Two Tailed Test If the alternative hypothesis specifies direction of the test, then one tailedOtherwise, two tailedMost cases
32 4. Calculating Test Statistics For one sample tests, use Z test statistic if population is Normal, is known, or if sample size is largeFor one sample tests, use T static if population distribution is not known or if sample size is small (less than 30)
33 5. Determine Degrees of Freedom Number of components that are free to vary about a parameterDf = Sample size – Number of parameters estimatedDf is n-1 for one sample test of mean
34 6. Compare the Computed Test Statistic Against a Tabled Value
35 Example of Testing Statistical Hypotheses About µ When σ is Known (Large Sample Test for Population Mean).
36 “Does Home Schooling Affect Educational Outcomes?” Research Question“Does Home Schooling Affect Educational Outcomes?”
37 Statistical Hypotheses Dr. Tate, a researcher at GMU decided to conduct a study to explore this question. He found out that every fourth-grade student attending school in Virginia takes CAT.Scores of CAT are normally distributed with µ = 250 and σ = 50.Home – schooled children are not required to take this test.
38 Statistical Hypotheses Dr. Tate selects a random sample of 36 home –schooled fourth graders and has each child complete the test. (It would be too expensive and time-consuming to test the entire population of home-schooled fourth-grade students in the sate.)Step 1: Specify HypothesesH0: µ = 250Ha: µ > 250α = 0.05
39 Calculated ZSelect the sample, calculate the necessary sample statisticsn=36σ =50
40 Critical Z Determine zα = 0.05 one sided CI of 95% Refer to the Z table and find the corresponding Z score:Z = 1.65
41 Make Decisions Regarding Ho Because the calculated z is greater than the critical z, Ho is rejected.1.80 > 1.65 and Ha is acceptedThe mean of the population of home-school fourth graders is not 250.
43 Using P value to Reject Hypothesis Step 3: Determine the p-value . A z of corresponds to a one tailed probability ofStep 4: Make decision regarding Ho. Because the p-value of is less than α =0.05H0 is rejected. The mean of the population of home-school fourth graders is not 250.
44 DECISION RULES In terms of z scores: If Zc > Zα Reject H0 In terms of p-value:If p value < α Reject H0
45 The One-sample Z TestOne-Sample tests of significance are used to compare a sample mean to a (hypothesized) population mean and determine how likely it is that the sample came from that population. We will determine the extent to which they occur by chance.We will compare the probability associated with our statistical results (i.e. probability of chance) with a predetermined alpha level.
46 The One-sample Z TestIf the probability is equal to or less than our alpha level, we will reject the null hypothesis and conclude that the difference is not due to chance.If the probability of chance is greater than our alpha level, we will retain the null hypothesis and conclude that difference is due to chance.
47 Take Home LessonProcedures for Hypothesis Testing and Use of These Procedures in One Sample Mean Test for Normal Distribution