Assignment 28
Psych 5500/6500
Fall, 2008
Chi-Square Test for the True Value of the Variance
1. Some researchers have been examining the effect of problem-solving training on
performance on a specific IQ test. Scoring of the test is such that normally (i.e. without
the training) the population will have a mean of 100 and a standard deviation of 10. The
researchers expect that the training will not only raise the mean score on the IQ test but
should also make the participants more similar in their scores. They sample some
students and give them the training, then they give them the IQ test. The data are
presented in the file ‘IQ.sav’. Analyze the data to test their theory that the training
decreases the variance of IQ scores.
a) Write H0 and HA
b) What is the expected (mean) value of chi-square if H0 is correct? (give a
numeric answer)
c) Based upon the story problem is this a directional or a non-directional theory
being tested?
d) Should the data be analyzed using a two-tail or a one-tail test?
I suggest you draw the sampling distribution and mark the rejection region and
the mean of the curve, but you don’t need to hand that in.
e) Use the Chi Square tool to arrive at the Chi Square critical value (note from
the image on the tool what rejection region it works with, then figure out what p
value you need to input to get the critical value you want).
f) Use SPSS to estimate the variance of the population of participants who have
received the problem solving training.
g) Compute the obtained value of chi-square.
h) Use the Chi Square tool to compute the value of p. Note from the image on
the tool what p value it gives you, and adjust accordingly.
i) What is your decision regarding H0? (it might help to draw the sampling
distribution and put in the rejection region)
j) What is your decision regarding whether or not you can conclude that the
training reduces variance of scores on the IQ test
k) This use of chi-square relies upon an assumption about the population, what is
that assumption?
Goodness of Fit Test
2. A researcher suspects that the ethnic makeup of an area has changed over some time
period. The last complete census (taken seven years ago) showed that the ethnic makeup
of the area was:
Ethnic Group A: 20%
Ethnic Group B: 30%
Ethnic Group C: 50%
The researcher randomly samples 39 people from the area and records to which ethnic
group they belong. The data can be found in the file “ethnicity”. Load the data into
SPSS.
a) Go to the Analyze>>DescriptiveStatistics>>Frequencies menu to get a
breakdown on the percentages of each ethnic group in the sample. Observe the
differences between the frequencies from 7 years ago and those of the more recent
survey. Note that if the more recent survey was a complete census (everyone in
the population being measured) then we would not need to do a statistical analysis
to get a p value, for we would have the actual values in the population and could
simply compare the data from then to now. But, that is not the case, the
researcher relied on a sample from the population and now we need to determine
whether the differences in percentages now is simply due to chance. State the
percent of each ethnic group in the sample.
b) To analyze the data using SPSS you will need to change the scores into
numbers. Use the ‘Recode into Different Variables’ to change ‘Group A’ to ‘0’,
‘Group B’ to ‘1’, and ‘Group C’ to ‘2’ (we did something like this earlier in the
semester). Look at your data to make sure it worked.
Now go to the Analyze>>NonparametericTests>>Chi-Square menu. Move the
numeric variable into the Test Variables List. Then enter the expected frequency
for each category (assuming H0 is true) in turn (first for 0, then for 1, then for 2)
into the Expected Values area. When finished click Ok. The ‘Asymp Sig’ is the p
value for this chi-square. State the results of the analysis in the following form:
χ²(df) = χ²obtained, p=...
c) Interpret the results of the analysis.
3. Let’s use the goodness of fit test to see if some data deviate significantly from what
we would expect if the data were normally distributed. Load into SPSS the crime data
from assignment 27. Take a look at a histogram of the Crime Rates and see that the data
look fairly skewed.
Step A: we need to change the crime rate data into standard scores. Go to the
Analyze>>DescriptiveStatistics>>Descriptives menu and move the Crime Rate variable
into the ‘Variables’ box, then click on the ‘Save standardized values as variables’ box,
and then click ‘OK’. Check to see that a new variable has been created that contains the
standard scores of the crime rates.
Step B: Now sort the z scores in descending order, that can be done through the
Data menu.
Step C: Create a new variable called ‘category’. Using the categories from the
lecture notes, give all the z scores of 1.15 and above a score of ‘1’, those between 0.68
and 1.15 a score of ‘2’, and so on. What if a score falls exactly on a boundary (e.g.
z=1.15)? This rarely happens, if it does, simply consistently follow some rule (e.g. if it
falls on the boundary assign it to upper category).
Step D: Go to the Analyze>>NonParametricTests>>ChiSquare, move the
‘category’ variable to the ‘Test Variable List’, under ‘Expected Values’ make sure ‘All
categories equal’ is selected (you are expecting each category to have the same frequency
if the data are normally distributed), and then click OK.
Step E: Under the ‘Test Statistics’ box the ‘Asymp Sig.’ is the p value for this
test.
a) Look at the expected values of the cells, do we need to be particularly worried
about whether or not the distribution of (O-E) is normally distributed? Why or
why not?
b) State the results of the analysis in the following form: χ²(df) = χ²obtained, p=...
c) Interpret the results of the analysis.
d) In general, if we reject H0 can we conclude that the data are not normally
distributed?
e) In general, if we fail to reject H0 can we conclude that the data are normally
distributed? Why or why not?
f) If the two highest crime rates were much higher, say with z scores of 4.2 and
3.9 respectfully (instead of 2.81 and 2.75 as they are in our data), then the data
would be much more skewed. Would this use of χ² to test the normality of the
data detect that change in the data?