Paired-Samples t Tests

Data Set

The same group of people rated both brands of coffee. Half of the subjects tasted brand A first. Half of the subjects tasted brand B first. 

Is there a difference in the mean ratings between the two brands of coffee?

The null hypothesis states that there is no difference in the mean ratings between the two brands of coffee. The difference between means equals zero.

The alternative hypothesis states that there is a significant difference in the mean ratings between the two brands of coffee. It is a two-sided test. 

Set a significance level

Use a .05 significant level.

SPSS for Windows

Input

Analysis

Choose Analyze \ Compare Means \ Paired-Samples T Test. Select both "a" and "b" as Paired Variables.

Click the right arrow button. Click OK. 

SPSS Output

• Paired Samples Statistics and Paired Samples Correlations

Note that the correlation between the two ratings scores are positive (.545). The high ratings on the brand A are associated with the high ratings on the brand B. However, the relationship is not significant (p > .05). 
 

• Paired Samples Test

Using the Traditional Formula to Compute the Paired-Samples t Test
 

• Compute the unbiased standard deviation (s) for each variable.
  
  
  
   
• Compute the correlation between the two variables
  
  Observations from the same subject are more likely to be similar than observations from different subjects.                                                                                

r = .545
 

• Unbiased standard deviation of the difference scores
  
  
  Compute the unbiased standard deviation for the difference scores. 

1. The unbiased standard deviation for the difference scores is equal to 1.059 as shown above.

2. You may also use the following formula to compute the unbiased standard deviation for the paired differences.

Since the same person rated both brands of coffee, the two ratings are correlated. The unbiased standard deviation for the paired differences is S(A-B) and can be also computed as 

 

S(A-B) = 1.059

Variance of a Difference: s _(A-B)

Note that there is a covariance  term (rs_As_B) since the two variables are related.

 

• Unbiased standard error of the mean difference
  
  The unbiased standard error of the mean difference is Sm(1-2) and can be computed as

 
 

• Now, we are ready to compute the t ratio.
   

 

The observed sample mean difference is .7. The standard error of the difference between two dependent means is .335. Compute the t ratio.

 

 

t = .7 / .335 = 2.09. 

 

• Locate the position of the t ratio of 2.09 in the t distribution with 9 degrees of freedom. It is a two-sided test.
  
  About 7% (.066) of the area under the t distribution lies beyond t ratios of +2.09 and -2.09.

The observed probability is larger than .05

 

Report the results.

A paired-samples t test was conducted to evaluate the hypothesis that there was a difference in the mean ratings between the two brands of coffee. The mean rating for coffee A was 7 (SD = 1.155) and the mean rating for coffee B was 6.3 (SD = 1.059). The test was not significant, t(9) = 2.09, p > .05.