Chapter 9 introduction to the t statistic
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Chapter 9 Introduction to the t-statistic. PSY295 Spring 2003 Summerfelt. Overview. CLT or Central Limit Theorem z-score Standard error t-score Degrees of freedom. Learning Objectives. Know when to use the t statistic for hypothesis testing Understand the relationship between z and t

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Chapter 9 introduction to the t statistic l.jpg

Chapter 9Introduction to the t-statistic

PSY295 Spring 2003

Summerfelt


Overview l.jpg
Overview

  • CLT or Central Limit Theorem

  • z-score

  • Standard error

  • t-score

  • Degrees of freedom


Learning objectives l.jpg
Learning Objectives

  • Know when to use the t statistic for hypothesis testing

  • Understand the relationship between z and t

  • Understand the concept of degrees of freedom and the t distribution

  • Perform calculations necessary to compute t statistic

    • Sample mean & variance

    • estimated standard error for X-bar


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Central limit theorem

  • Based on probability theory

  • Two steps

    • Take a given population and draw random samples again and again

    • 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

  • Even if the population distribution is skewed, the distribution from Step 2 will be normal!


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Z-score Review

  • A sample mean (X-bar) approximates a population mean (μ)

  • The standard error provides a measure of

    • how well a sample mean approximates the population mean

    • determines how much difference between X-bar and μ is reasonable to expect just by chance

  • The z-score is a statistic used to quantify this inference

  • obtained difference between data and hypothesis/standard distance expected by chance


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What’s the problem with z?

  • Need to know the population mean and variance!!! Not always available.


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What is the t statistic?

  • “Cousin” of the z statistic that does not require the population mean (μ) or variance (σ2)to be known

  • Can 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


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When to use the t statistic?

  • For single samples/groups,

    • Whether a treatment causes a change in the population mean

    • Sample mean consistent with hypothesized population mean

  • For two samples,

    • Coming later!


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Difference between X-bar and μ

  • Whenever you draw a sample and observe

    • there is a discrepancy or “error” between the population mean and the sample mean

    • difference between sample mean and population

  • Called “Sampling Error” or “Standard error of the mean”

  • Goal for hypothesis testing is to evaluate the significance of discrepancy between X-bar & μ


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Hypothesis Testing Two Alternatives

  • Is the discrepancy simply due to chance?

    • X-bar = μ

    • Sample mean approximates the population mean

  • Is the discrepancy more than would be expected by chance?

    • X-bar ≠μ

    • The sample mean is different the population mean


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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 variance

    • Or you can use standard deviation



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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 statistics

    • Describes how scatted the scores are around the mean

    • Divide by n-1 or df

  • Estimated standard error is a inferential statistic

    • measures how accurately the sample mean describes the population mean

    • Divide by n


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The t statistic

  • The 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 error

  • What’s the difference between the t formula and the z-score formula?


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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 z

  • In smaller samples, the estimation is poorer

  • Why?

  • Degrees of freedom is used to describe how well t represents z


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Degrees of freedom

  • df = n – 1

  • Value of df will determine how well the distribution of t approximates a normal one

    • With larger df’s, the distribution of the t statistic will approximate the normal curve

    • With smaller df’s, the distribution of t will be flatter and more spread out

  • t table uses critical values and incorporates df


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Four step procedure for Hypothesis Testing

  • Same procedure used with z scores

  • State hypotheses and select a value for α

    • Null hypothesis always state a specific value for μ

  • Locate a critical region

    • Find value for df and use the t distribution table

  • Calculate the test statistic

    • Make sure that you are using the correct table

  • Make a decision

    • Reject or “fail to reject” null hypothesis


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Example

  • GNC is selling a memory booster, should you use it?

  • Construct a sample (n=25) & take it for 4 weeks

  • Give sample a memory test where μ is known to be 56

  • Sample produced a mean of 59 with SS of 2400

  • Use α=0.05

  • What statistic will you use? Why?


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Steps

  • State Hypotheses and alpha level

  • Locate critical region (need to know n, df, & α)

  • Obtain the data and compute test statistic

  • Make decision