New Pension Plan

Date: 07/15/99 at 17:58:17
From: la tara mcgraw
Subject: Math - statistics

A sample of the employees of Carter industries regarding acceptance of 
a new pension plan revealed the following:

opinion regarding new pension plan

age     superior   very good   good    fair    unsatisfactory
20-29      19         27        25      52          81
30-39      10         17        15      29          41
40-49      51         40        31      21          27
50-       142         81        16      9            8

Is there a relation between age and an employee's opinion of the new 
plan? Test the .05 level.

Answer: Chi-square problem?

Date: 07/15/99 at 19:21:16
From: Doctor Anthony
Subject: Re: Math - statistics

Table of Observed Frequencies.

   Age  | Superior  V.good  Good   Fair   Unsatis.| Total
  ------|-----------------------------------------|---------
  20-29 |    19       27     25     52       81   | 204             
  30-39 |    10       17     15     29       41   | 112
  40-49 |    51       40     31     21       27   | 170
  50-   |   142       81     16      6        8   | 253
 -------|-----------------------------------------|--------
        |   222      165     87    108      157   | 739 


Table of Expected Values

   Age  | Superior  V.good  Good   Fair   Unsatis.| Total
  ------|-----------------------------------------|---------
  20-29 |    61       46     24     30       43   | 204             
  30-39 |    34       25     13     16       24   | 112
  40-49 |    51       38     20     25       36   | 170
  50-   |    76       56     30     37       54   | 253
 -------|-----------------------------------------|--------
        |   222      165     87    108      157   | 739 

This is a 5 x 4 table so degrees of freedom = 4 x 3 = 12

The calculation of chi-squared is given in the next table

     O     E     |O-E|     |O-E|^2     |O-E|^2/E
   -----------------------------------------------
     19    61                             28.9
     27    46                              7.8
     25    24                              0.04
     52    30                             16.1
     81    43                             33.6 
     10    34                             16.9
     17    25                              2.6
     15    13                              0.31
     29    16                             10.6
     41    24                             12.0
     51    51                              0.0
     40    38                              0.10
     31    20                              6.0
     21    25                              0.64
     27    36                              2.25
    142    76                             57.3
     81    56                             11.2
     16    30                              6.5
      6    37                             26.0 
      8    54                             39.2
   --------------------------------------------------    
                            chi-squared = 278.04

And the tabular value of X^2 5% (degrees of freedom = 12) is 21.03

So our value is VERY significant and we reject the null hypothesis 
that the opinion is independent of age.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/