Hypothesis testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are t-tests, which are tests that examine whether two means are statistically significantly different from each other or whether the difference between them simply occurred by chance. A One-Sample T-Test compares a sample mean to a known population mean. An Independent Samples T-Test compares two sample means from different populations regarding the same variable. A Paired Samples T-Test compares two sample means from the same population regarding the same variable at two different times such as during a pre-test and post-test, or it compares two sample means from different populations whose members have been matched.

### One-Sample T-Test

A One-Sample T-Test compares a sample mean and a known population mean to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the respondents' number of children with the known 2013 United States fertility rate of 2.06 children per woman. The One-Sample T-Test is examining whether the difference between the sample mean number of children per respondent is significantly different from the known population fertility rate.

To generate a One-Sample T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'One-Sample T Test...' in the side menu.

In the One-Sample T Test dialog box that pops up, select the variable of interest (Number of Children, childs) from the list of variables and bring it over to the 'Test Variable(s)' field. Then, enter the known population mean in the 'Test Value:' field. Click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of two tables. The first table, One-Sample Statistics, contains statistical information about the Number of Children variable, such as N, the Mean, the Standard Deviation, and the Standard Error of the Mean. The second table, One-Sample Test, contains information specific to the One-Sample T-Test, such as the Test Value, the t value, the df (degrees of freedom), the alpha 2-tailed Significance value (when the Sig. value is .05 or less, the probability that the difference between the sample mean and the test value was due to chance is 5% or less), the Mean difference (the difference between the sample mean and the test value), and a 95% Confidence Interval of the Difference. In this case, the difference between the sample mean number of children (1.89) and the known population mean number of children (2.06) is significant.

### Independent-Samples T-Test

An Independent-Samples T-Test compares two sample means from different populations regarding the same variable to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the mean number of hours spent emailing per week (Email Hours Per Week, emailhr) by married respondents and single respondents (Not Married, absingle). The 'Email Hours Per Week, emailhr' variable is the test variable, and the 'Not Married, absingle' variable is the nominal grouping variable. The Independent-Samples T-Test is examining whether the difference between the mean number of hours married respondents spent emailing and the mean number of hours single respondents spent emailing is significantly different or occurred by chance.

To generate a Independent-Samples T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'Independent-Samples T Test...' in the side menu.

In the Independent-Samples T Test dialog box that pops up, select the variable of interest (Email Hours Per Week, emailhr) from the list of variables and bring it over to the 'Test Variable(s)' field. Then, enter the nominal grouping variable (Not Married, absingle) in the 'Grouping Variable:' field. Then click 'Define Groups...' to identify the two populations being compared.

In the Define Groups dialog box, select 'Use specifiec values' and enter the Value Labels of the nominal grouped variable. In this example, Group 1 will correspond with Value Label 1, which refers to the respondents who indicated they are not married ('Yes'), and Group 2 will correspond with Value Label 2, which refers to the respondents who indicated they are married ('No'). Then, click 'Continue,' and back in the Independent-Samples T Test dialog box, click OK.

The output is displayed in the SPSS Viewer window. The output consists of two tables. The first table, Group Statistics, contains statistical information about the Email Hours Per Week variable, split by whether the respondent is not married (indicated in the chart by Yes) or is married (indicated in the chart by No). For each group of respondents, the N, the Mean, the Standard Deviation, and the Standard Error of the Mean are displayed.

The second table, Independent-Samples Test, contains information specific to the Independent-Samples T-Test, such as information about Levene’s Test for Equality of Variances and the t-test for Equality of Means. Levene’s Test for Equality of Variances tests whether variability within each group (married or not married) is equal. The outputs displays two sets of results: one set in which equal variance is assumed and one set in which equal variance is not assumed. It is up to the user to determine which set of results is appropriate. You can determine which results are appropriate by looking at the Sig., the alpha level of significance. If the alpha level is greater than .05, then group variances are assumed to be equal. In this example, Sig. is greater than .05 so group variances are assumed to be equal, and we read the top line of the table. The portion of the table dedicated to the t test for Equality of Means displays the t value, the df (degrees of freedom), the 2-tailed Sig. value (when the Sig. value is .05 or less, the probability that the difference between the two means was due to chance is 5% or less), the Mean difference (the difference between the two means), and a 95% Confidence Interval of the Difference. In this case, the difference between the two means is not significant and could have occured by chance.

### Paired Samples T-Test

A Paired Samples T-Test compares two sample means from the same population regarding the same variable at two different times such as during a pre-test and post-test, or it compares two sample means from different populations whose members have been matched, to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the respondents' mean number of children (Number of Children, childs) with the respondents' mean ideal number of children (Ideal Number of Children, chldidel). The respondents in this example are paired with themselves. The Paired-Samples T-Test is examining whether the difference between the mean number of children and the mean ideal number of children is significantly different or occurred by chance.

To generate a Paired-Samples T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'Paired-Samples T Test...' in the side menu.

In the Paired-Samples T Test dialog box that pops up, select the variables of interest (Number of Children, childs, and Ideal Number of Children, chldidel) from the list of variables and bring them over to the 'Paired Variables:' field. Then, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of three tables. The first table, Paired Samples Statistics, contains statistical information about the Number of Children and Ideal Number of Children variables. For each variable, the Mean, the N, the Standard Deviation, and the Standard Error of the Mean are displayed. The second table, Paired Samples Correlations, contains a correlation value measuring how closely related the two variables are to each other. The correlation value is the correlation coefficient of the two variables and measures the strength and direction of the linear relationship between the two variables. Specifically, the closer the correlation value to 1 or -1, the more strongly linearly related the variables. In this example, the correlation value is not close to 1 so the variables do not have a strong linear relationship.

The third table, Paired Samples Test, displays the Mean (referring to the difference between the two means), the Standard Deviation of the Mean, the Standard Error of the Mean, and a 95% Confidence Interval of the Difference. The table also displays the t value, the df (degrees of freedom), and the 2-tailed Sig. value (when the Sig. value is .05 or less, the probability that the difference between the two means was due to chance is 5% or less). In this case, the difference between the two means is significant.

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