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Chapter 9 Introduction to the t-statistic PSY295 Spring 2003 Summerfelt

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Overview o CLT or Central Limit Theorem o z-score o Standard error o t-score o Degrees of freedom

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Learning Objectives o Know when to use the t statistic for hypothesis testing o Understand the relationship between z and t o Understand the concept of degrees of freedom and the t distribution o Perform calculations necessary to compute t statistic o Sample mean & variance o estimated standard error for X-bar

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Central limit theorem o Based on probability theory o Two steps 1. Take a given population and draw random samples again and again 2. Plot 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 error o Even if the population distribution is skewed, the distribution from Step 2 will be normal!

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Z-score Review o A sample mean (X-bar) approximates a population mean (μ) o The standard error provides a measure of o how well a sample mean approximates the population mean o determines how much difference between X-bar and μ is reasonable to expect just by chance o The z-score is a statistic used to quantify this inference o obtained difference between data and hypothesis/standard distance expected by chance

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What’s the problem with z? o Need to know the population mean and variance!!! Not always available.

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What is the t statistic? o “Cousin” of the z statistic that does not require the population mean (μ) or variance (σ 2 )to be known o Can be used to test hypotheses about a completely unknown population (when the only information about the population comes from the sample) o Required: a sample and a reasonable hypothesis about the population mean (μ) o Can be used with one sample or to compare two samples

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When to use the t statistic? o For single samples/groups, o Whether a treatment causes a change in the population mean o Sample mean consistent with hypothesized population mean o For two samples, o Coming later!

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Difference between X-bar and μ o Whenever you draw a sample and observe o there is a discrepancy or “error” between the population mean and the sample mean o difference between sample mean and population o Called “Sampling Error” or “Standard error of the mean” o Goal for hypothesis testing is to evaluate the significance of discrepancy between X-bar & μ

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Hypothesis Testing Two Alternatives o Is the discrepancy simply due to chance? o X-bar = μ o Sample mean approximates the population mean o Is the discrepancy more than would be expected by chance? o X-bar ≠ μ o The sample mean is different the population mean

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Standard error of the mean o In Chapter 8, we calculated the standard error precisely because we had the population parameters. o For the t statistic, o We use sample data to compute an “Estimated Standard Error of the Mean” o Uses the exact same formula but substitutes the sample variance for the unknown population variance o Or you can use standard deviation

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Estimated standard error of mean Or

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Common confusion to avoid o Formula for sample variance and for estimated standard error (is the denominator n or n-1?) o Sample variance and standard deviation are descriptive statistics o Describes how scatted the scores are around the mean o Divide by n-1 or df o Estimated standard error is a inferential statistic o measures how accurately the sample mean describes the population mean o Divide by n

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The t statistic o The t statistic is used to test hypotheses about an unknown population mean (μ) in situations where the value of (σ 2 ) is unknown. o T=obtained difference/standard error o What’s the difference between the t formula and the z-score formula?

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t and z o Think of t as an estimated z score o Estimation is due to the unknown population variance (σ 2 ) o With large samples, the estimation is good and the t statistic is very close to z o In smaller samples, the estimation is poorer o Why? o Degrees of freedom is used to describe how well t represents z

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Degrees of freedom o df = n – 1 o Value of df will determine how well the distribution of t approximates a normal one o With larger df’s, the distribution of the t statistic will approximate the normal curve o With smaller df’s, the distribution of t will be flatter and more spread out o t table uses critical values and incorporates df

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Four step procedure for Hypothesis Testing o Same procedure used with z scores 1. State hypotheses and select a value for α o Null hypothesis always state a specific value for μ 2. Locate a critical region o Find value for df and use the t distribution table 3. Calculate the test statistic o Make sure that you are using the correct table 4. Make a decision o Reject or “fail to reject” null hypothesis

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Example o GNC is selling a memory booster, should you use it? o Construct a sample (n=25) & take it for 4 weeks o Give sample a memory test where μ is known to be 56 o Sample produced a mean of 59 with SS of 2400 o Use α=0.05 o What statistic will you use? Why?

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Steps 1. State Hypotheses and alpha level 2. Locate critical region (need to know n, df, & α) 3. Obtain the data and compute test statistic 4. Make decision

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