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 Deviation SD 2 ( X M ) N Population Standard Deviation s 2 ( X M ) ( N 1) Calculating the Estimated Population SD > Step 1: Calculate the sample mean > Step 2: Use the sample mean in the corrected standard deviation formula s (X M ) ( N 1) 2 Steps to calculating s: (8 12 16 12 14) M 12.4 5 s 2 ( X M ) ( N 1) 35.2 (5 1) = 8.8 = 2.97 Calculating Standard Error for the t Statistic > Using the standard error s SM N > The t statistic (M M ) t SM Steps to calculating t statistic using standard error: > From previous example: s 2 ( X M ) ( N 1) = 2.97 SM s 2.97 1.33 N 5 > Assume population mean is 11: ( M M ) (12.4 11) t 1.05 SM 1.33 The t Statistic > When sample size increases, s approaches σ and t and z become more equal > The t distributions • Distributions of differences between means 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 Example: Single Sample t Test 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? Calculating the Single Sample t Test STEP 2: State the hypotheses H0 : 1 = 2 H1: 1 2 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 t Test Calculation Completed STEP 5: Calculate the test statistic ( M M ) (7.8 4.6) t 2.873 SM 1.114 STEP 6: Make a decision Calculating Confidence Intervals > Draw a picture of the distribution > Indicate the bounds > Look up the t statistic > Convert the t value into a raw mean 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% Confidence Interval Example STEP 4: Convert the t statistics back into raw means. 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 (M ) d s 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 Stop and Think > 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.