Was the Dependent Test Legitimate?

Scenario:
In a previous project, a jury was given the task of determining if female managers earned less than their male counterparts.  The jury was presented with the salaries of 40 managers (20 male and 20 female). 

In an effort to discredit labor's defense, management argues labor used an illegitimate test and therefore their conclusion is invalid.  A prerequisite for hypothesis testing of dependent (paired) data requires that the sample size be large (at least 30) OR the differences are Normally distributed and contains no outliers.

Project:
I.  Normal Probability Plot
     A.  Find the differences for the paired data (Female, L1 - Male managers, L2).  Store the differences in L3 and
           fill in the table.

Differences: Female - Male 

    B.  Use the Normal Probability Plot on your calculator to subjectively determine if the differences are Normally
          distributed.  Make a copy of your TI-83 / 84 screen and include it with your project.

Using the TI-83 or TI-84 series calculator to determine if the graph of the Normal Probability Plot is linear and the data set is Normally distributed.

1.  Order the data in ascending order. 2.  Select STAT and enter the data into list 1 (L1). 3.  Press "2nd" "Y=" to access "STAT PLOT" 4.  Select "1: Plot 1...Off" by pressing "ENTER".

5.  Select "ON" and press "ENTER" 6.  Use the arrows to select "Normal Probabilty Plot" icon.  It is the third      figure in the second row of "Type" and press "ENTER". 7.  Use the down arrow and enter "L1" into "DATA  List:" 8.  Use the down arrow and highlight "X" in "Data Axis" and        press "ENTER". 9.  Use the down arrow and highlight the symbol you want for the data.

10.  Select "Zoom". 11. Select "9: ZoomStat".

The calculator runs the Normal Probability Plot program and graphs the data on the x axis and their corresponding z-scores on the y axis.   The differences do appear linear.  The differences are Normally distributed.

II. Make a boxplot of the differences to determine if there are any outliers.  Include a copy of the boxplot with
      your project.

Using the TI-83 to calculate an IQR boxplot.

1.  To use the TI-83 to construct an IQR boxplot       a.  Press "2nd", "Y = " and "ENTER".  This will take you to the            "Stat Plots" Menu       b.  Under "Plots1"             i.  Highlight "ON" and press "ENTER".             ii.  Use the arrows to highlight the IQR Boxplot and press "ENTER".             iii.  Use the arrows to get to "XList": and  press "2nd" "3".                     This refers the calculator to the data stored in List 3.             iv. Enter "1" at "Freq:" and press "ENTER".            v.  Select the mark you want to identify any outlers.

2.  Using the same "Window" settings for the histogram, press "GRAPH" Note:  The square to the right of the IQR boxplot means that there is at least one value that is an outlier.

III.  Based on your decisions, should the jury believe Management or Labor?  Explain your decision.
        The jury should believe Labor's claim that the female managers earn less than their male
        counterparts.  Labor's use of dependent hypothesis testing was legitimate.  The sample of
        differences is normally distributed.

Note:  Did the plot of the data look linear?  This determination of the linearity of the line is "subjective".   Statistics is based on "objective" tests of the data.    However, a statistical software package is needed to objectively determine if the data is normally distributed.  Using only the TI-83 calculator, we have to be subjective in making this determination.

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