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One-Sample t Test
In this example 83 University of Dayton students took the UCLA Loneliness Scale, version 3. The loneliness scale has a population mean of 40. Higher values on the scale reflect greater loneliness. The researcher wants to know if UD students are less lonely than people in general.
- Step 1: Write the null and alternative hypotheses and specify the probability of making a Type I error:
H 0 : μ ≥ 40 H 1 : μ < 40
α =.
- Step 2: Sketch and determine the critical region:
df = n – 1 = 83 – 1 = 82 From a table of critical t scores with α= .05, one-tailed, tcritical(82) = -1. (The critical t is negative because the critical region is concentrated in the left tail of the distribution.)
- Step 3: Calculate the test statistic:
a. Open SPSS
b. Either type the data or open a data set. The class data set is available from http://academic.udayton.edu/gregelvers/psy216/SPSS/loneliness.sav. Save the
-6 -4 -2 0 2 4 6
Critical Region
data file somewhere and open it with SPSS.
c. Analyze | Compare Means | One-Sample T Test (this means to click on the Analyze menu item, then click on the Compare Means option in the drop down menu and then click on the One-Sample T Test option from the menu.)
d. Move the dependent variable (the variable measured by the researcher, loneliness) into the Test Variable(s) box. You can either drag the variable name into the box, or select the variable by clicking on it and then clicking on the arrow button between the two boxes.
e. Click in the Test Value box and type the value specified in the null hypothesis (40 in this example).
This gives us the degrees of freedom, 82 (df = N – 1 = 83 – 1), the value of t, -2.586 (t = ( - μ) / sm = (37.506 – 40) / 0.96430 = -2.586), the 95% confidence interval of the difference for a two tailed test , from -4.4123 to -0.5757, and the p value for a two tailed test = .011.
- Step 4: Make a decision: Notice that SPSS gave the p value for a two tailed hypothesis, but our hypothesis is one tailed (directional). Divide the two tailed p value by 2 to get the one tailed p value. The one tailed p value is .011 / 2 = .006 (rounded from .0055).
If the p value is less than or equal to the α level, then you should reject H 0. Otherwise, you should fail to reject H 0. Because p = .006 and α = .05, we reject H 0 and conclude that it is likely the case that UD students are less lonely than the population in general.
- Cohen’s d must be calculated by hand, but you can get both of the values used in the formula from the SPSS output:
Estimated value of Cohen’s d = mean difference / standard deviation = -2.49398 / 8.78521 = -0.28.
This is a small effect (between .2 and .5)
- r^2 must be calculated by hand. Again, both values for the formula can be gotten from the SPSS output:
r^2 = t^2 / (t^2 + df) = -2.586^2 / (-2.586^2 + 82) =.
This is a small effect (between .01 and .09).
- The 95% confidence interval for the mean is calculated from:
± tcritical ∙ sm = 37.506 ± 1.664 ∙ 0.96430 = 34.297 to 39.
