One-Sample t-Test: Analyzing Sample Means vs. Population

CHAPTER 5 The One-Sample tTest T he one-sample t test is used when the mean of one sample is compared to some known or estimated population mean. The sample is typically different from the population in some way, and the question of interest is whether the mean of the sample differs significantly from the mean of the pop ulation on a dependent variable of interest. An example of a one-sample t test is presented below. Example An industrial psychologist hired by a leading accounting firm wanted to know if the average num ber of hours worked per week at the firm was significantly different from the national average of 52 hours for accountants. The industrial psychologist randomly sampled 16 people from different divi sions within the firm and recorded the average number of hours they worked per week over a three month period. The dependent variable in this study is the number of hours worked per week. Objective and Data Requirements of the One-Sample t Test The One-Sample t Test Objective Data Requirements Example To determine whether the mean of a sample differs significantly from some known or estimated population mean. One sample of participants Accountants at leading accounting firm Dependent variable Dependent variable Continuous • Number of hours worked per week Null and Alternative Hypotheses The null hypothesis (HQ) states that the number of hours worked per week at the leading account ing firm is equal to the national average of 52 hours: H 0 - fx = 52 The alternative hypothesis (H\) states that the number of hours worked per week at the leading accounting firm is not equal to the national average of 52 hours: ^ : ix * 52 62

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Chapter 5 / The One-Sample t Test 63 Evaluation of the NuLL Hypothesis The one-sample t test provides a test of the null hypothesis that the number of hours worked per week at the accounting firm is equal to 52 hours. If the t test produces results that seem unlikely if the null hypothesis is true (results that occur less than 5% of the time), then the null hypothe sis is rejected. If the t test produces results that seem fairly likely if the null hypothesis is true (results that occur greater than 5% of the time), then the null hypothesis is not rejected. Research Question The fundamental question of interest in a research study can also be expressed in the form of a research question, such as, "Is the number of hours worked per week for employees at a leading accounting firm different from the national average of 52 hours?" The Data The data for the 16 participants are presented in Figure 5.1. Participant 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Hours worked per week 54 48 68 53 60 45 57 62 71 60 55 63 68 64 56 60 Figure 5.1 The number of hours worked per week by the 16 employees at the leading accounting firm. (Note: The participant variable is included for illustration but will not be entered into SPSS.) Data Entry and Analysis in SPSS Steps 1 and 2 below describe how to enter the data in SPSS. The data file is also on the web at www.pearsonhighered.com/yockey under the name hours worked.sav in the Chapter 5 folder. If you prefer to open the file from the web site, skip to Step 3. Step 1: Create the Variable 1. Start SPSS. 2. Click the Variable View tab. In SPSS, one variable will be created for the number of hours worked per week by the employees at the leading accounting firm. The variable will be named hoursweek. 64 Unit II / Inferential Statistics

3. Enter the name hoursweek and press the down arrow key (I). See Figure 5.2 for details. • Untitled! [DataSetO] - PASW Statist! Data Trartstorfn Analyze Graphs LBiiies Adverts V*WGW Help n MI MM,J ^ 1. I hoursweek Numeric : gataView variable View ; h^ m ml Vidth Decimals Label Values Missing : Columns I Align 2 None :Nona 8 B Riant PASWStaii.fti':: erccessor BTeatfy : Role \ Input

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Figure 5.2 The Variable View window with the variable hoursweek entered. Step 2: Enter the Data 1. Click the Data View tab. The variable hoursweek appears in the first column of the Data View window. 2. To enter the data for hoursweek, make sure the first cell in the upper left-hand corner of the window is active (if it isn't active, then click on it). Enter the number of hours worked for each of the participants, beginning with the first participant's value (54) and continuing until the last participant's value (60) has been entered. (After entering the value for a participant, press either the down-arrow key ( i ) or the Enter button to move to the next row in the Data View window.) The completed data file in presented in Figure 5.3. I I "Untitled! [DataSetO] Fil Ec Vie Da Trans Anal Gra| Utilii AM- Wine He | i S« 18: hoursweek m ^ Visible: 1 of 1 ' Variables 1 3 4 5 : 6 J 7 8 9 10 11 12 13 J 14 15 HZ hoursweek 54.00 48.00 68.00 53.00 60.00 45.00; 57.00= 62.00 71.00 60.00 55.00 63.00 68.00 : 64.00 56.00 60.00 *-!: ^•- I •*• I Data View : Variable View :: cessans rea..

Figure 5.3 The completed data file for the one-sample t test example. Chapter 5 / The One-Sample t Test 65 Step 3: Analyze the Data 1. From the menu bar, select Analyze > Compare Means > One-Sample T Test ... (see Figure 5.4). I t *Untitled1 [DataSetO] PASW Statistics Data Editor .zM3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 Data Tram 54.00 48.00 68.00 53.00 60.00 45.00 57.00 62.00 71 00 60.00 55.00 63 00 68.00 64.00 56.00 60 00 :form Analyze Graphs Utilities ,. J Reports Descriptive Statistics Tables Coggers telesfts General Linear Model Generalised Linear Mode Mixed Models Correlate Regression Loglinear Classify Dimension Reduction Scale Nonparametric Tests Forecasting Survival Muflipte Response . :mg Value Analysis. Multiple Imputation Complex Samples Quality Control Add-ons Window : $MtM M me-Sample T Test.. | ^ Paired-Samples T Test.. . :-i?-WayANOVA...

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Oats View . * ! :v - :> Figure 5.4 Menu commands for the one-sample t test. 2. A One-Sample T Test dialog box appears with the variable hoursweek on the left hand side of the dialog box (see Figure 5.5). I l l One-Sample T Test Test Variable(s) Sptior Test Value: 0 ieset Cancel , Hell Figure 5.5 The One-Sample T Test dialog box. 3. Select the dependent variable, hoursweek, and click the right-arrow button O* ) to move it into the Test Variable(s) box. In the Test Value box, enter 52. This is the value that is specified in the null hypoth esis. See Figure 5.6 (page 66) for details. [Note: This step is critical and is often overlooked. If the default value of zero is left in the Test Value box, the one-sample t would test whether the employees at the leading accounting firm work significantly 66 Unit II / Inferential Statistics One-Sample T Test lest Vaiiable(s): ursweek OK Paste Reset Cancel Help

Test Value: 52 The value specified in the null hypothesis (52) goes into the Test Value box. Figure 5.6 The One-Sample T Test dialog box (continued). different from zero hours per week (which they certainly should!). Be sure to always enter the value specified in the null hypothesis (in this case, 52) into the Test Value box when performing a one-sample t test in SPSS.] 5. Click OK. The one-sample t test procedure runs in SPSS and the results are presented in the Viewer window. Step 4: Interpret the Results The output of the one-sample t test is displayed in Figure 5.7. One-Sample Statistics hoursweek N 16 Mean 59.0000 Std. Deviation 7.14609 One-Sample Test Std. Error Mean 1.78652 The Mean Difference of 7.00 is equal to the difference between the mean number of hours worked per week at the leading accounting firm (59), and the value of 52 (the population for accountants) specified in the null hypothesis.

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Test Value = 52 / / Mean 95% Confidence Interval of the Difference

hoursweek t 3.918 df 15 Sig. (2-tailed) .001„ Difference 7.00000 Lower 3.1921 Upper 10.8079

Since the p-value of .001 is less than .05, the null hypothesis is rejected. Figure 5.7 Output for the one-sample t test. One-Sample Statistics Table The first table of output, One-Sample Statistics, displays the sample size (N), mean, standard devi ation, and standard error of the mean for the sample. Notice that the average number of hours worked per week at the leading accounting firm is 59, which is 7 hours more than the national aver age of 52. Whether this difference (of 7 hours) is large enough to be statistically significant will be considered next. One-Sample Test Table The next table, One-Sample Test, provides the answer to our research question, that is, whether the average number of hours worked at a leading accounting firm is different from the national average of 52 hours. The test of the null hypothesis is provided in the form of a t, where: Chapter 5 / The One-Sample t Test 67 Difference between the mean of the sample and the mean of the population t = Standard error of the mean

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Substituting the appropriate values (Mean Difference, Std. Error Mean) from the One-Sample Test table yields a t value of t = = 3.918 1.78652 which agrees with the value provided in the One-Sample Test table in column t. With 16 partici pants, the degrees of freedom (df) are equal to 15 (number of participants —1), with a corre sponding /j-value of .001. Because the p- value of .001 is less than .05, the null hypothesis that the average number of hours worked per week at the leading accounting firm is equal to 52 is rejected. Since the test is significant, inspecting the mean (once again) for the 16 participants from the One-Sample Statistics table shows that the average number of hours worked at the accounting firm (59) was significantly higher than the national average of 52. Effect Sizes As was described in the Unit II introduction, hypothesis testing indicates whether or not there is a significant difference between the groups (answering a yes/no question), while effect sizes pro vide an indication of the magnitude of the results (answering a "To what degree?" question). In the current example, with hypothesis testing we concluded that there was a difference in the number of hours worked between the employees at the leading accounting firm and the national average. An effect size will now be calculated to express how large that difference is. A commonly used effect size statistic for the one-sample t test is Cohen's d, where Difference between the mean of the sample and the mean of the population d = Standard deviation Substituting the appropriate values from the One-Sample Test table in Figure 5.7 (Mean Difference, Std. Deviation) yields a d of Jacob Cohen (1988), a pioneer in the development of effect size measures, provided guidelines for what constitutes small, medium, and large effect sizes in practice. While the guidelines provided by Cohen are in widespread use (and will also be used in this text), it should be noted that they are only approximations (a sentiment stressed by Cohen) and can vary considerably across disciplines. Cohen's guidelines for small, medium, and large effect sizes for the one-sample t test are .20, .50, and .80, respectively. These values indicate the amount of difference between the sample mean and the population mean in terms of standard deviation units. Therefore, a value of .20 indi cates one-fifth of a standard deviation difference between the groups, .50 indicates one-half of a standard deviation difference, and .80 indicates eight-tenths of a standard deviation difference. Using Cohen's conventions, a d of .98 corresponds to a large effect in practice and indicates a difference of almost 1 standard deviation in the number of hours worked between the employees at the leading accounting firm and the national average for accountants (with the employees at the leading accounting firm working more). 1 Expression of the Results in APA Format For each of the chapters in Unit II, a write-up of the results will be provided using the format of the American Psychological Association (APA). Specific guidelines for APA format may be found in the Publication Manual of the American Psychological Association (2009). 2 68 Unit II / Inferential Statistics In writing the results, the conclusion of the hypothesis test, the degrees of freedom, the rvalue, the /?-value, and effect size are reported along with the mean and standard deviation of the sample. An example of a brief write-up in APA format is presented next. Written Results Employees at a leading accounting firm (M = 59.00, SD = 7.15) work significantly more hours per week than the national average of 52 hours, f(15) = 3.92, p < .05, d = 0.98. Assumptions of the One-Sample t Test For each of the chapters in Unit II, the assumptions of the statistical procedures will be described. Assumptions are important because, if they are not met, the results of a given statistical procedure can be untrustworthy (i.e., the /j-values can be inaccurate). Whether a test is compromised by an assumption violation, however, depends both on the specific assumption that is violated (some assumptions are much worse than others to violate) and the degree to which the assumption is not met. The assumptions of the one-sample t test are described next.

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1. The observations are independent. This assumption should be satisfied by designing your study so the participants do not influence each other in any way. Violating this assumption can seriously compro mise the accuracy of the one-sample t test. If there is reason to believe the indepen dence assumption has been violated, the one-sample t test should not be used. 2. The dependent variable is normally distributed in the population. This assumption means that the number of hours worked per week should be nor mally distributed in the population. For moderate to large samples sizes (N > 30), most types of nonnormal distributions tend to have relatively little impact on the accuracy of the t test, although some nonnormal distributions can adversely affect the power of the t test. One remedy for nonnormal data is to perform a nonlinear trans formation on the data; nonlinear transformations, however, are beyond the scope of this text [the interested reader is referred to Tabachnick and Fidell (2007) for more information on nonlinear transformations]. Summary of Steps for Conducting a One-Sample tTest in SPSS I. Data Entry and Analysis 1. Create one variable in SPSS. 2. Enter the data. 3. Select Analyze > Compare Means > One-Sample T Test 4. Move the dependent variable to the Test Variable(s) box. 5. In the Test Value box, enter the value specified in the null hypothesis. 6. Click OK II. Interpretation of the Results 1. Check the p- value (reported as "sig." (2-tailed)) in the One-Sample Test table. • If p < .05, the null hypothesis is rejected. Compare the mean of the sample to the mean of the population and write the results indicating whether the sample mean is greater than or less than the population mean. • If p > .05, the null hypothesis is not rejected. Write the results indicating that there is not a significant difference between the mean of the sample and the mean of the population. Chapter 5 / The One-Sample r Test 69 Exercises For a research project, a student wanted to test whether people who claim to be successful at picking winning teams in football are able to select "winners" at dif ferent than chance levels (i.e., picking winners more or less than 50% of the time). She identifies 15 people who advertise their ability to pick "winners" and records the percentage of correct picks for each person over an entire football season. The percentage of correct picks for the 15 "prognosticators" is presented in Figure 5.8. Prognosticates 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Percentage of correct picks 45 46 47 52 51 43

47 38 53 51 52 50 48 47 51

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Figure 5.8 Percentage of correct picks for the 15 prognosticators. Enter the data in SPSS and perform the appropriate analyses to answer the questions below (name the variable success). (Hint: For this problem, use a Test Value of 50 and do not include a 7c sign when you enter the data.) a. State the null and alternative hypotheses. b. State a research question for the data. c. Do the prognosticators pick winners at a rate different from 50%? Test at a = .05. d. What is the effect size? Would you characterize the effect size as small, medium, or large' 1 e. Write the results of the study using APA format as appropriate. The mean score on a standardized math skills test is known to be 50 for the entire U.S. population of fourth graders (with higher scores indicating better perfor mance). A new math skills training program was being used at a local school dis trict, and an administrator was charged with the task of evaluating whether the children under the new program have performance that is different from the national average (preferably the performance is higher than the national average, but test for either possibility). Twenty-five fourth graders who were instructed using the math skills program were administered the standardized math exam. The exam scores for the students are provided in the file Chapter 5_Exercise l.sav in the Chapter 5 folder online at www.pearsonhighered.com/yockey (the variable is named mathscore). Open the file in SPSS and perform the appropriate analyses to answer the questions below. a. State the null and alternative hypotheses. b. State a research question for the data. c. Are the scores on the math exam for the fourth graders different from the national average? Test at a = .05. d. What is the effect size? Would you characterize the effect size as small, medium, or large? e. Write the results of the study using APA format as appropriate.

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