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 Sample Standard DeviationPopulation 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

= 8.8 = 2.97 Steps to calculating s:

>Using the standard error >The t statistic Calculating Standard Error for the t Statistic

= 2.97 Steps to calculating t statistic using standard error: >From previous example: >Assume population mean is 11:

>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 session Population 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 STEP 3: Determine the characteristics of 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

>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: = and = 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 For the counseling center data:

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