Chapter 7

Introduction to the t Test

Part 1: One-sample t test

Sept. 25, 2014

t Test for a Single Sample• Z test requires that you know from pop• Use a t-test when you don’t know the

population standard deviation.• One sample t-test:– Compare a sample mean to a population with a

known mean but an unknown variance

Hypothesis testing procedure for t-test• Same general procedure:– Assume null hypothesis is true, relative to an alternate

(research hypothesis)– Compute observed t statistic from sample data based

on sampling distribution of the mean– Determine cutoff point (now a critical t) in comparison

distribution based on – Reject the null hypothesis if your observed t value falls

in critical region • |t observed| > |t critical|

One–sample t Test• Must estimate the population variance from the

sample scores

– Unbiased estimate of the population variance (S2)

11

)( 22

N

SS

N

MXS

Sum of squareddeviations

Use N-1 to makeS2 bigger than sample

t Test for a Single Sample

• Degrees of freedom– Number of scores that are

“free to vary”– Formula for S2 using degrees of freedom

1Ndf

df

SS

df

MXS

2

2 )(

Note: S2 indicates we estimated this from a sample ( always indicates population info)

t Test for a Single Sample• Also need to find standard deviation of

distribution of means (SM)• The variance of the distribution of means

• The standard deviation of the distribution of means

22

N

SSM

2MM SS

t Test for a Single Sample

• For t-tests, relevant comparison distribution is t distribution (not the normal curve used before in z test)

• The t distribution

One-sample t Test• State null & research hypotheses• Assuming null is true, compute observed t

statistic for your sample mean

• Find correct critical t value based on your df• When do we reject the null?

MS

Mt

Note on t-table• When > 30 df, critical values only given for 35, 40,

45 df, etc.• If your df are in between these groups, be more

conservative and use the lower df

• Do example in class…

One Sample t-test in SPSS• Use menus for:

Analyze Compare Means One sample tGives pop-up menu…need 2 things:– select variable to be tested/compared to

population mean (use “self-esteem”)– Notice “test value” window at bottom. Enter the

population/comparison mean here (use x.xx from past semester)

– Hit OK, get output and find sample mean, observed t, df, significance value (if < .05, reject null)