7.3.1. Do two processes have the same mean?

7. Product and Process Comparisons
7.3. Comparisons based on data from two processes

7.3.1. Do two processes have the same mean?

Testing hypotheses related to standard deviations from two processes

Given two random samples of measurements,

from two independent processes, there are three types of questions regarding the true standard deviations of the processes that can be addressed with the sample data. They are:

Are the means from the two processes the same?
Is the mean of one process less than the other?
Is the mean of one process greater than the other?

Typical null hypotheses

The corresponding null hypotheses that test the true mean of the first process,

, against the true mean of the second process,

are:

H₀: =
H₀: <
H₀: >

Basic statistics from the two processes

The basic statistics for the test are the sample means

;

and the sample standard deviations

;

with degrees of freedom and respectively.

Form of the test statistic where the two processes have equivalent standard deviations

If the standard deviations from the two processes are equivalent, and this should be tested before this assumption is made, the test statistic is

where the pooled standard deviation is estimated as

with degrees of freedom .

Form of the test statistic where the two processes do NOT have equivalent standard deviations

If it cannot be assumed that the standard deviations from the two processes are equivalent, the test statistic is

The degrees of freedom are not known exactly but can be estimated using the Welch-Satterthwaite approximation

Test strategies

The strategy for testing the hypotheses under (1), (2) or (3) above is to calculate the appropriate t statistic from one of the formulas above, and then perform a test at significance level

, where

is chosen to be small, typically .01, .05 or .10. The hypothesis associated with each case enumerated above is rejected if:

Explanation of critical values

The critical values from the t table depend on the significance level and the degrees of freedom in the standard deviation. For hypothesis (1)

is the

upper critical value from the t table with

degrees of freedom and similarly for hypotheses (2) and (3).

Example of unequal number of data points

A new procedure to assemble a device is introduced and tested for possible improvement in time of assembly. The question being addressed is whether the mean,

, of the new assembly process is better than the mean,

, for the old assembly process as in hypothesis (3). Data (in minutes required to assemble a device) for both the old and new processes are listed below along with their relevant statistics.


        Device    Process 1    Process 2

           1            32            35
           2            37            31
           3            35            29
           4            28            25
           5            41            34
           6            44            40
           7            35            27
           8            31            32
           9            34            31
          10            38
          11            42


Mean                36.0909        32.2222
Standard deviation   4.9082         2.5874
No. measurements         11              9
Degrees freedom          10              8

Computation of the test statistic

From this table we generate the test statistic

with the degrees of freedom approximated by

Decision process

For a one-sided test at the 5% significance level, go to the t table for 5% signficance level, and look up the critical value for degrees of freedom

= 16. The critical value is 1.746. Thus, hypothesis (3) cannot be rejected because the test statistic (t = 1.713) is greater than -1.746 and, therefore, we conclude that process 2 has improved assembly time (smaller mean) over process 1.