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 9Introduction 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
• 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
• 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
• 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?
• Need to know the population mean and variance!!! Not always available.
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?
• 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 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
• 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
• 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
• 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 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
• 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
• 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
• 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
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?
Steps
• State Hypotheses and alpha level
• Locate critical region (need to know n, df, & α)
• Obtain the data and compute test statistic
• Make decision