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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

Chapter 9Introduction to the t-statistic

PSY295 Spring 2003


  • CLT or Central Limit Theorem
  • z-score
  • Standard error
  • t-score
  • Degrees of freedom
learning objectives
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
central limit theorem
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!
z score review
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
what s the problem with z
What’s the problem with z?
  • Need to know the population mean and variance!!! Not always available.
what is the t statistic
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
when to use the t statistic
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!
difference between x bar and
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
hypothesis testing two alternatives
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
standard error of the mean
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
common confusion to avoid
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
the t statistic
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?
t and z
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
degrees of freedom
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
four step procedure for hypothesis testing
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
  • 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?
  • State Hypotheses and alpha level
  • Locate critical region (need to know n, df, & α)
  • Obtain the data and compute test statistic
  • Make decision