Step 4: Make a Decision and Report the Results
III. Two-Sample t Test: Repeated Measures Design
Research Question. A study is being conducted at a sleep disorders clinic to determine if a regimen of swimming exercises affects the sleep patterns of individuals with insomnia.
The number of hours of nightly sleep of 26 insomniac patients is recorded for a two-week period. The patients are then exposed to two weeks of daily swimming exercises, and their sleep is monitored for two more weeks. For this sample of patients, the amount of nightly sleep increased by MD = 1.2 hours with SSD = 112.
a. On the basis of this data, did the swimming regimen affect the amount of sleep? Use a two-tailed test with α = .05 and determine the effect size.
b. Make a point estimate of how much sleep increases on average as a result of the swimming regimen.
c. Establish a 95% confidence around the mean difference and write a sentence of interpretation.
d. Establish a 99% confidence around the mean difference and write a sentence of interpretation.
A. Step 1: Formulate Hypotheses
Step 2: Indicate the Alpha Level and Determine Critical Values
(Continued)
Step 3: Calculate Relevant Statistics
Step 4: Make a Decision and Report the Results
Effect size
B. Point Estimate
C. 95% Confidence Interval
Interpretation
D. 99% Confidence Interval
Interpretation
Your Turn!
(continued)
As was apparent in the last “Your Turn!” practice exercise, when we increase the level of con- fidence, we lose specificity. The range of values becomes broader and is thus more likely to include the true population parameters. Conversely, when we reduce the level of confidence, we gain specificity but lose confidence in our estimation. This is illustrated in the normal distribution below.
The 95% confidence level establishes a narrower (and more specific) interval of values, while the 99% level provides a wider (and less specific) interval of values.
Answers to “Your Turn!” Problems I. Confidence Interval for a One-Sample t Test
A. M = 72
B. LL = 68.94 and UL = 75.06
We can be 99% confident that the population mean would be between 68.94 and 75.06.
II. Two-Sample t Test: Independent Samples Design
A. Step 1: Step 2: Step 3:
H0: μ1 = μ2 α = .05 SS1 = 154.86 H1: μ1 ≠ μ2 df = 11 SS2 = 112.83
tcrit = ± 2.201 sM M1 2 2 75. tobt = +3.28
(Continued)
Step 4:
Subjects who heard positive remarks about taking the CAT scored significantly higher than the group who heard no such remarks. Reject H0, t(11) = +3.28, p < .05.
Effect size: d = 1.83, large effect B. M1 − M2 = 9.03
C. LL = 2.98 and UL = 15.08
We can be 95% confident that the difference between population means would be between 2.98 and 15.08.
III. Two-Sample t Test: Repeated Measures Design
A. Step 1: Step 2: Step 3:
H0: μD = 0 α = .05 sD = 2.12 H1: μD ≠ 0 df = 25 SMD =.42
tcrit = ± 2.060 tobt = +2.86 Step 4:
The swimming regimen had a significant effect on sleep. Reject H0, t(25) = +2.86, p < .05.
Effect size: d = .57, moderate effect B. MD = 1.2
C. LL = 1.2 – (2.060)(.42) and UL = 1.2 + (2.060)(.42)
= .33 = 2.07
We can be 95% confident that the increased amount of sleep will be between .33 and 2.07 hours.
D. LL = 1.2 – (2.787)(.42) and UL = 1.2 + (2.787)(.42)
= .03 = 2.37
We can be 99% confident that the increased amount of sleep will be between .03 and 2.37 hours.
Answers to “Your Turn!” Problems (continued)
Using Microsoft Excel for Data Analysis
if you are using excel for the first time for statistical analysis, you may need to load the add-in tool that allows these functions. the information for loading the data analysis toolPak as well as general instructions for using excel for data analysis are at the beginning of the excel section in Chapter 4.
Confidence Interval for a One-Sample t Test
this chapter looked at confidence intervals for three different kinds of t tests. to calculate these confidence intervals, we will be providing the formulas and instruct- ing excel to grab the values needed in the formulas from the cells where they are located the spreadsheet. We’ll start by using excel to calculate a confidence interval for the one-sample t test using the example from your text about finger dexterity exercises for typing speed. the formula for the lower limit of the confidence interval for a one-sample t test is: LL = M − t(SM). the formula for the upper limit is: UL = M + t(SM).
first, type in the information shown below in column a of the spreadsheet that will be needed for the calculations. also, type in the summary information in column C that was provided for this problem. We will now instruct excel to perform the remain- ing operations:
1. Critical t-values. rather than using the table at the back of your text, excel will determine your tcrit values. With your cursor in cell C4, go to the Formulas tab and click on the Insert Function (fx) command from the ribbon. from the Insert Function dialog box, scroll down and highlight T.INV.2T. this is the function that will provide the tcrit values for a two-tailed test. Click OK.
since we are establishing a 90% confidence interval, we will enter .10 in the probability window of the Function Arguments box as well as the appropriate df (16 in this case). after clicking OK, the tcrit values needed for the confidence interval formulas (1.745884) will appear in the cell.
2. Sample Standard Deviation. to determine the standard error of the mean needed for the confidence interval formulas, you will first have to obtain the sample stan- dard deviation. With your cursor in cell C5, type the following exactly as written:
= sQrt(C2/(C3–1)) and hit the enter key. the sample standard deviation (15) will appear in the cell.