Chapter 8

HYPOTHESIS TESTING USING THE ONE-SAMPLE t-TEST

Going Forward

Your goals in this chapter are to learn:• The difference between the z-test and the t-

test• How the t-distribution and degrees of

freedom are used• When and how to perform the t-test• What is meant by the confidence interval for

, and how it is computed

Using the t-Test

• Use the z-test when is known

• Use the t-test when is not known and must be estimated by calculating

X

Xs

X

Understanding the One-Sample t-Test

Setting Up the Statistical Test

1. Set up the statistical hypotheses (H0 and Ha) in

precisely the same fashion as in the z-test

2. Select alpha

3. Check the assumptions for a t-test

Assumptions for a t-Test

• You have a one-sample experiment using interval or ratio scores

• The raw score population forms a normal distribution

• The variability of the raw score population is estimated from the sample

Performing the One-Sample t-Test

Performing the One-Sample t-Test

1. Compute the estimated population variance (s ) using the formula2Xs

1

)( 22

2

NNX

XsX

Performing the One-Sample t-Test

2. Compute the estimated standard error of the mean ( ) using the formulaXS

N

ss XX

Performing the One-Sample t-Test

3. Calculate the tobt statistic using the formula

Xs

Xt

obt

The t-Distribution

The t-distribution is the distribution of all possible values of t computed for random sample means selected from the raw score population described by H0

Comparison of Two t-Distributions Based on Different Sample Ns

Degrees of Freedom

• The quantity N – 1 is called the degrees of freedom (df )

• This is the number of scores in a sample that reflect the variability in the population

Using the t-Table

Obtain the appropriate value of tcrit from the t-tables using•The correct table depending on whether you are conducting a one-tailed or a two-tailed test,•The appropriate column for the chosen and •The row associated with your degrees of freedom (df)

Interpreting the t-Test

A Two-Tailed t-Distribution for df = 8When H0 is True and = 10

Reaching a Decision

If tobt is beyond tcrit in the tail of the distribution:

1.Reject H0 ; accept Ha

2.Conclude there is a relationship between your independent variable and dependent variable

3.Describe the relationship

Reaching a Decision

If tobt is not beyond tcrit :

1.Fail to reject H0

2.Consider if your power level was sufficient3.Conclude you have no evidence of a

relationship between your independent variable and dependent variable

One-Tailed Tests

If you believe your sample represents a population where the mean is greater than some value (e.g., 25):

H0: ≤ 25

Ha: > 25

One-Tailed Tests

If you believe your sample represents a population where the mean is less than some value (e.g., 25):

H0: ≥ 25

Ha: < 25

Summary of the One-Sample t-Test

1. Create the two-tailed or the one-tailed H0 and Ha

2. Compute tobt

1. Compute and2. Compute3. Compute tobt

3. Create the sampling t-distribution and use df = N – 1 to find tcrit in the t-tables

4. Compare tobt to tcrit

X 2Xs

Xs

Estimating by Computing a Confidence Interval

Estimating

There are two ways to estimate the population mean (

Point estimation in which we describe a point on the dependent variable at which the population mean () is expected to fall

Interval estimation in which we specify a range of values within which we expect the population mean () to fall

Confidence Intervals

• We perform interval estimation by creating a confidence interval

• The confidence interval for a single describes an interval containing values of

XtsXts XX ))(())(( critcrit

Example

Use the following data set and conduct a two-tailed t-test to determine if = 12

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

• • Choose = 0.05• Reject H0 if tobt > +2.110 or if tobt < -2.110

• Since 4.4 > 2.110, reject H0 and conclude does not equal 12

12:;12:0 aHH

40.4380.0

67.1

18

61.11267.13

obt

N

sX

tX