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

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

3. 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

4. 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!

5. 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

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 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

8. 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!

9. 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

10. 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

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 variance
Or you can use standard deviation

12. Estimated standard error of mean

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 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

14. 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?

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 z
In smaller samples, the estimation is poorer
Why?
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 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

17. 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

18. 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?

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