Presentation on theme: "Chapter 9 Introduction to the t-statistic"— Presentation transcript:
1 Chapter 9 Introduction to the t-statistic PSY295 Spring 2003Summerfelt
2 Overview CLT or Central Limit Theorem z-score Standard error t-score Degrees of freedom
3 Learning ObjectivesKnow when to use the t statistic for hypothesis testingUnderstand the relationship between z and tUnderstand the concept of degrees of freedom and the t distributionPerform calculations necessary to compute t statisticSample mean & varianceestimated standard error for X-bar
4 Central limit theorem Based on probability theory Two steps Take a given population and draw random samples again and againPlot the means from the results of Step 1 and it will be a normal curve where the center of the curve is the mean and the variation represents the standard errorEven if the population distribution is skewed, the distribution from Step 2 will be normal!
5 Z-score ReviewA sample mean (X-bar) approximates a population mean (μ)The standard error provides a measure ofhow well a sample mean approximates the population meandetermines how much difference between X-bar and μ is reasonable to expect just by chanceThe z-score is a statistic used to quantify this inferenceobtained difference between data and hypothesis/standard distance expected by chance
6 What’s the problem with z? Need to know the population mean and variance!!! Not always available.
7 What is the t statistic?“Cousin” of the z statistic that does not require the population mean (μ) or variance (σ2)to be knownCan be used to test hypotheses about a completely unknown population (when the only information about the population comes from the sample)Required: a sample and a reasonable hypothesis about the population mean (μ)Can be used with one sample or to compare two samples
8 When to use the t statistic? For single samples/groups,Whether a treatment causes a change in the population meanSample mean consistent with hypothesized population meanFor two samples,Coming later!
9 Difference between X-bar and μ Whenever you draw a sample and observethere is a discrepancy or “error” between the population mean and the sample meandifference between sample mean and populationCalled “Sampling Error” or “Standard error of the mean”Goal for hypothesis testing is to evaluate the significance of discrepancy between X-bar & μ
10 Hypothesis Testing Two Alternatives Is the discrepancy simply due to chance?X-bar = μSample mean approximates the population meanIs the discrepancy more than would be expected by chance?X-bar ≠ μThe sample mean is different the population mean
11 Standard error of the mean In Chapter 8, we calculated the standard error precisely because we had the population parameters.For the t statistic,We use sample data to compute an “Estimated Standard Error of the Mean”Uses the exact same formula but substitutes the sample variance for the unknown population varianceOr you can use standard deviation
13 Common confusion to avoid Formula for sample variance and for estimated standard error (is the denominator n or n-1?)Sample variance and standard deviation are descriptive statisticsDescribes how scatted the scores are around the meanDivide by n-1 or dfEstimated standard error is a inferential statisticmeasures how accurately the sample mean describes the population meanDivide by n
14 The t statisticThe t statistic is used to test hypotheses about an unknown population mean (μ) in situations where the value of (σ2) is unknown.T=obtained difference/standard errorWhat’s the difference between the t formula and the z-score formula?
15 t and z Think of t as an estimated z score Estimation is due to the unknown population variance (σ2)With large samples, the estimation is good and the t statistic is very close to zIn smaller samples, the estimation is poorerWhy?Degrees of freedom is used to describe how well t represents z
16 Degrees of freedom df = n – 1 Value of df will determine how well the distribution of t approximates a normal oneWith larger df’s, the distribution of the t statistic will approximate the normal curveWith smaller df’s, the distribution of t will be flatter and more spread outt table uses critical values and incorporates df
17 Four step procedure for Hypothesis Testing Same procedure used with z scoresState hypotheses and select a value for αNull hypothesis always state a specific value for μLocate a critical regionFind value for df and use the t distribution tableCalculate the test statisticMake sure that you are using the correct tableMake a decisionReject or “fail to reject” null hypothesis
18 Example GNC is selling a memory booster, should you use it? Construct a sample (n=25) & take it for 4 weeksGive sample a memory test where μ is known to be 56Sample produced a mean of 59 with SS of 2400Use α=0.05What statistic will you use? Why?
19 Steps State Hypotheses and alpha level Locate critical region (need to know n, df, & α)Obtain the data and compute test statisticMake decision