**Case 1:
Independent Samples & Equal Variances
Case 2:
Independent Samples & Unequal Variances
Case 3:
Dependent Samples: Matched/Paired Observations
**

**Case 1:
Assumptions, Problem Description & Data, and
Discussion of the Results.
I. Underlying Assumptions**

1. The samples (n

2. One or both sample sizes are less than 30

3. The appropriate sampling distribution of the test statistic is the t distribution

4. The unknown variances of the two populations are equal

Women who are union members earn $2.50 per hour more than women who are not union members. (

Union Workers (n

22.40 | 18.90 | 16.70 | 14.05 | 16.20 | 20.00 | 16.10 | 16.30 | 19.10 | 16.50 |

18.50 | 19.80 | 17.00 | 14.30 | 17.20 |

Non-union Workers (n_{2}=
20):

17.60 | 14.40 | 16.60 | 15.00 | 17.65 | 15.00 | 17.55 | 13.30 | 11.20 | 15.90 |

19.20 | 11.85 | 16.65 | 15.20 | 15.30 | 17.00 | 15.10 | 14.30 | 13.90 | 14.50 |

**Question**:
Does there appear to be any difference in the mean wage rate between these
groups?

**Data Entry
**Note that HOURLY WAGES of female workers in
manufacturing is the variable of interest (X); it is a continuous variable. To enter the
values, double-click on

(1) The assumption of equal population variance σ

(2) Unlike the

A. The Outputs/Results

Union Membership | N | Mean | Std. Deviation | Std. Error Mean | |
---|---|---|---|---|---|

Hourly Wages of Female Workers | union | 15 | 17.5367 | 2.2403 | .5784 |

non-union | 20 | 15.3600 | 1.9885 | .4446 |

Levene's Test for Equality of Variances | t-test for Equality of Means | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | Std. Error Difference | 95% Confidence Interval of the Difference | |||

Lower | Upper | |||||||||

Hourly Wages of Female Workers | Equal variances assumed | .411 | .526 | 3.036 | 33 | .005 | 2.1767 | .7169 | .7180 | 3.6353 |

Equal variances not assumed | 2.983 | 28.183 | .006 | 2.1767 | .7296 | .6826 | 3.6707 |

The first output table contains
summary statistics for the two groups. The second table contains all the statistics that
are needed to perform the test. Notice that SPSS/win actually performs the test under two
alternative assumptions about the population variance -- equal and unequal variance.
Because equal variance is assumed in this example we will use the results in that row for
the ensuing discussion.

**B. Testing Procedure
**I will summarize the testing procedure under

Ho

Note that this is a t-test because the distribution of hourly wages in the populations of unionized and non-unionized women is assumed to be normal and also both

__ Decision Rules__: Reject

From the "Group Statistics" table the pooled sample variance can be derived as

S

Because this is a one-tailed test (to the right of the sampling and/or 't' distribution), alpha = .05 must not be divided by 2 when determining the value of the

**Case 2:
Assumptions, Problem Description & Data, and
Discussion of the Results.**

**I. Underlying
Assumptions**

1. The samples (n_{1} and n_{2}) from two normal populations are independent

2. One or both sample sizes are less than 30

3. The appropriate sampling distribution of the test statistic is the t distribution

4. The unknown variances of the two populations are not equal

II. Problem Description and
Data

Starting annual salary for individuals entering the
public accounting and financial planning professions were presented in *Fortune*,
June 26, 1995. The starting salaries for a sample of 12 public accountants and 14
financial planners are below. Data are in thousands of dollars (ASW, 1998, p. 403).

Public Accountant (n_{1} = 12)

30.6 | 31.2 | 28.9 | 35.2 | 25.1 | 33.2 | 31.3 | 35.3 | 31.0 | 30.1 | 29.9 | 24.4 |

Financial Planner (n_{2} = 14)

31.6 | 26.6 | 25.5 | 25.0 | 25.9 | 32.9 | 26.9 | 25.8 | 27.5 | 29.6 | 23.9 | 26.9 | 24.4 | 25.5 |

**Question:**

Using alpha = .05, test for any difference between
the population mean starting annual salaries for the two professions. What is your
conclusion?

**
Data Entry
**Note that the starting annual SALARY for persons
entering public accounting and financial planning professions is the variable of interest
(X); it is a continuous variable. To enter the values, double-click on

(1) The assumption of unequal population variance (i.e., σ

(

A. The Outputs/Results

profession | N | Mean | Std. Deviation | Std. Error Mean | |
---|---|---|---|---|---|

Starting Salary of Public Accountant & Financial Planner | Public Accountant | 12 | 30.517 | 3.347 | .966 |

Financial Planner | 14 | 27.000 | 2.641 | .706 |

Levene's Test for Equality of Variances | t-test for Equality of Means | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

F | Sig. | t | df | Sig. (2-tailed) | Mean Difference | Std. Error Difference | 95% Confidence Interval of the Difference | |||

Lower | Upper | |||||||||

Starting Salary of Public Accountant & Financial Planner | Equal variances assumed | .292 | .594 | 2.994 | 24 | .006 | 3.517 | 1.175 | 1.093 | 5.941 |

Equal variances not assumed | 2.939 | 20.848 | .008 | 3.517 | 1.197 | 1.027 | 6.006 |

The first output table contains
summary statistics for the two groups. The second table contains all the statistics that
are needed to perform the test. As in the Case 1 above, SPSS/win actually performs the
test under two alternative assumptions about the population variance -- equal and unequal
variance. Because unequal variance is assumed in this example we will use the
results in that row for the ensuing discussion.

**B. Testing Procedure
**Again, I will summarize the testing procedure
under

Ho

Note that this is a t-test because the distribution of salaries in the populations of public accounting and financial planning professions is assumed to be normal and also both

From the "Group Statistics" table the sample variance of the public accountants' salaries is given as S

This case considers a research situation in which two samples are not independent. This situation occurs when each individual observation (i) within a sample is related (matched or paired) to an individual observation in the second sample. The relatedness may be the result of the individual observations in the two samples

1. representing before and after results (which is presented in this example),

2. having matching characteristics,

3. being matched by location, or

4. being matched by time.

If there are definite reasons for pairing (or matching) the individual observations in the two samples, the two samples are dependent rather than being independent. Generally, the precision from an analysis of dependent samples is greater than that from the analysis of independent samples. Thus, if paired analysis is appropriate, it is the preferred approach.

Client |
Before | After |

1 | 140 | 132 |

2 | 160 | 158 |

3 | 210 | 195 |

4 | 148 | 152 |

5 | 190 | 180 |

6 | 170 | 164 |

**Question**:
Using alpha = .05, test to determine whether the introductory program provides
a

statistically significant weight loss. What is
your conclusion?

**
Data Entry
**Note that the

(1) The key to the analysis of the matched/paired design sample design is to realize that we consider only the column of differences d

A. The Outputs/Results

Mean | N | Std. Deviation | Std. Error Mean | ||
---|---|---|---|---|---|

Pair 1 | Client's Weight After | 163.50 | 6 | 22.00 | 8.98 |

Client's Weight Before | 169.67 | 6 | 26.39 | 10.78 |

N | Correlation | Sig. | ||
---|---|---|---|---|

Pair 1 | Client's Weight After & Client's Weight Before | 6 | .979 | .001 |

Paired Differences | t | df | Sig. (2-tailed) | ||||||
---|---|---|---|---|---|---|---|---|---|

Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||

Lower | Upper | ||||||||

Pair 1 | Client's Weight After - Client's Weight Before | -6.17 | 6.59 | 2.69 | -13.08 | .74 | -2.294 | 5 | .070 |

The first output table contains
summary statistics for the paired samples. The second table reports the Pearson
correlation coefficient for weight before and after the program. The value of .979
suggests the existence of a strong positive association between the clients weight before
and after the program; the p-value of .001 indicates that the observed association is
statistically significant at 5% level. The third table contains all the statistics that
are needed to perform the test.

**
B. Testing Procedure
**Again, I will summarize the testing procedure
under

Ho

the weight after is greater than the weight before)

of the program)

Note that this is a t-test because the distribution of weight in the population of all possible clients is assumed to be normal; also the sample size 'n' is small (i.e., n < 30). Thus, the test statistic

From the "Paired Samples test" table we can obtain the following summary statistics: the mean weight loss after introduction of the program is

**Copyright© 1996, Ebenge Usip, all rights reserved.
Last revised: Wednesday, July 24, 2013.**