1. Hypothesis Testing Using The One-Sample t-Test
Chapter 8
Hypothesis Testing Using
The One-Sample t-Test

2. 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 m, and how it is computed

3. 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

4. Understanding the One-Sample t-Test

5. Setting Up the Statistical Test
Set up the statistical hypotheses (H0 and Ha) in precisely the same fashion as in the z-test
Select alpha
Check the assumptions for a t-test

6. 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

7. Performing the One-Sample t-Test

8. Performing the One-Sample t-Test
Compute the estimated population variance (s ) using the formula

9. Performing the One-Sample t-Test
Compute the estimated standard error of the mean ( ) using the formula

10. Performing the One-Sample t-Test
Calculate the tobt statistic using the formula

11. 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

12. Comparison of Two t-Distributions Based on Different Sample Ns

13. 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

14. 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 a, and
The row associated with your degrees of freedom (df)

15. Interpreting the t-Test

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

17. Reaching a Decision
If tobt is beyond tcrit in the tail of the distribution:
Reject H0 ; accept Ha
Conclude there is a relationship between your independent variable and dependent variable
Describe the relationship

18. Reaching a Decision If tobt is not beyond tcrit : Fail to reject H0
Consider if your power level was sufficient
Conclude you have no evidence of a relationship between your independent variable and dependent variable

19. One-Tailed Tests
If you believe your sample represents a population where the mean is greater than some value (e.g., 25):
H0: m ≤ 25
Ha: m > 25

20. One-Tailed Tests
If you believe your sample represents a population where the mean is less than some value (e.g., 25):
H0: m ≥ 25
Ha: m < 25

21. Summary of the One-Sample t-Test
Create the two-tailed or the one-tailed H0 and Ha
Compute tobt
Compute and
Compute
Create the sampling t-distribution and use df = N – 1 to find tcrit in the t-tables
Compare tobt to tcrit

22. Estimating m by Computing a Confidence Interval

23. Estimating m There are two ways to estimate the population mean (m)
Point estimation in which we describe a point on the dependent variable at which the population mean (m) is expected to fall
Interval estimation in which we specify a range of values within which we expect the population mean (m) to fall

24. Confidence Intervals
We perform interval estimation by creating a confidence interval
The confidence interval for a single m describes an interval containing values of m

25. Example
Use the following data set and conduct a two-tailed t-test to determine if m = 12 14 13 15 11 10 12 17

26. Example
Choose a = 0.05
Reject H0 if tobt > or if tobt <
Since 4.4 > 2.110, reject H0 and conclude m does not equal 12