Hypothesis Testing Using The One-Sample t-Test 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 m, 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

Understanding the One-Sample t-Test

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

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 Compute the estimated population variance (s ) using the formula

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

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

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

Interpreting the t-Test

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

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

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

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

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

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

Estimating m by Computing a Confidence Interval

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

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

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

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