The Single-Sample t Test

Chapter 9

The t Distributions

> Distributions of Means When the Parameters Are Not Known

> Using t distributions • Estimating a population standard deviation

from a sample

N

MXSD

2)(

)1(

)( 2

N

MXs

Sample Standard Deviation

Population Standard Deviation

Calculating the Estimated Population SD

> Step 1: Calculate the sample mean

> Step 2: Use the sample mean in the corrected standard deviation formula

)1(

)( 2

N

MXs

= 8.8 = 2.97

Steps to calculating s:

)1(

)( 2

N

MXs )15(

2.35

(8 12 16 12 14)12.4

5M

> Using the standard error

> The t statistic

Calculating Standard Error for the t Statistic

N

sSM

M

M

S

Mt

)(

= 2.97

Steps to calculating t statistic using standard error:

)1(

)( 2

N

MXs

> From previous example:

> Assume population mean is 11:

2.971.33

5M

sS

N

( ) (12.4 11)1.05

1.33M

M

Mt

S

> When sample size increases, s approaches σ and t and z become more equal

> The t distributions• Distributions of differences between means

The t Statistic

Wider and Flatter t Distributions

Check Your Learning

> When would you use a z test? Give an example.

> When would you use a t test? Give an example.

Hypothesis Tests: The Single Sample t Test

> The single sample t test • When we know the population mean, but

not the standard deviation• Degrees of freedom

df = N - 1 where N is sample size

Stop and think. Which is more conservative: one-tailed or two-tailed tests? Why?

> The t test• The six steps of hypothesis testing

> 1. Identify population, distributions, assumptions> 2. State the hypotheses> 3. Characteristics of the comparison distribution> 4. Identify critical values

df =N-1

> 5. Calculate> 6. Decide

STEP 1: Identify population, distribution, assumptions

Population 1: All clients at this counseling center who sign a contract to attend at least 10 sessionPopulation 2: All clients at this counseling center who do not sign a contract to attend at least 10 sessions

• The comparison distribution will be a distribution of means

• Use a single-sample t test because there is one sample and we know the population mean but not the population standard deviation

• Assumptions?

Example: Single Sample t Test

Calculating the Single Sample t Test

STEP 2: State the hypotheses

0 1 2

1 1 2

H : =

H :

STEP 3: Determine the characteristicsof the comparison distribution.

t Test Calculation Continued

STEP 4: Determine the critical values, or cutoffs

df = N -1 = 5 -1 = 4

STEP 5: Calculate the test statistic

STEP 6: Make a decision

t Test Calculation Completed

( ) (7.8 4.6)2.873

1.114M

M

Mt

S

> Draw a picture of the distribution> Indicate the bounds> Look up the t statistic> Convert the t value into a raw mean

Calculating Confidence Intervals

Example Confidence Interval

STEP 1: Draw a picture of a t distribution that includes the confidence interval

STEP 2: Indicate the bounds of the confidence interval on the drawing

Confidence Interval Continued

STEP 3: Look up the t statistics that fall at each line marking the middle 95%

STEP 4: Convert the t statistics back into raw means.

Confidence Interval Example

Confidence Interval Completed

STEP 5: Check that the confidence interval makes sense

The sample mean should fall exactly in the middle of the two ends of the interval:

4.71-7.8 = -3.09 and 10.89 - 7.8 = 3.09

The confidence interval ranges from 3.09 below the sample mean to 3.09 above the sample mean.

Interpretation of Confidence Interval

If we were to sample five students from the same population over and over, the 95% confidence interval would include the population mean 95% of the time.

Calculating Effect size

s

Md

)(

For the counseling center data:

(M ) (7.8 4.6)d 1.29

s 2.490

Dot Plots

> The dot plot is a graph that displays all the data points in a sample, with the range of scores along the x-axis and a dot for each data point above the appropriate value.

> Dot plots serve a similar function to stem-and-leaf plots.

> The three steps to creating a dot plot

STEP 1: We determine the lowest score and highest score of the sample

STEP 2: We draw an x-axis and label it, including the values from the lowest through highest scores

STEP 3: We place a dot above the appropriate value for every score.

Example Dot Plot

> When would you use a z test over a t test?

> When would you use an independent sample t test? Think of a specific study.

Stop and Think