STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

Week 11 STAT170 workshop on CATEGORICAL variables. Prepared for
Numeracy Centre Macquarie University, by Nan Carter.
Examples taken from various sources and adapted for STAT170.
CATEGORICAL VARIABLES: WE CONSIDER ONLY ONE RANDOM SAMPLE IN
STAT170
SUMMARY:
1. ONE CATEGORICAL VARIABLE OBSERVED ON EACH ITEM
• TWO CATEGORIES: use EITHER
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z-TEST OF PROPORTIONS FOR H0: π = π0 where π0 has a
particular value, and
provided that nπ > 5 and n(1-π) > 5
z = (p - π0 )/SE(p) where p is the sample estimate of π
and SE(p)= √[(π)(1-π)/n]
The 95% confidence interval for π uses the
estimated SE(p) = √[(p)(1-p)/n]
and is given by (p – 1.96 SE(p), p + 1.96 SE(p))
OR χ 2 GOODNESS OF FIT TEST:
The same null hypothesis as for the z-test, and it
is used to find expected values, E, for both
categories. The observed COUNTS are denoted by ‘O’.
The χ 2 -test is valid if all expected values, E, are
>5;
χ 2 = Σ[(O-E) 2 /E] with 1 degree of freedom
• MORE THAN TWO CATEGORIES: use only the χ 2
goodness of fit test with degrees of freedom =
(no. of categories–1)
Use info from the question to form the null
hypothesis, and thence the expected values. The
observed counts are given in the data.
WARNING: SMALL EXPECTED FREQUENCIES: The chi-square
test is not valid when any expected frequency is less
than 5. To counter this problem, we group classes
together to create larger observed counts and larger
expected counts until the problem goes away! There
is a good example of the need to do this in the 1998
Nan Carter: workshop notes prepared for Numeracy Centre Macquarie University.