Chi- square
goodness of fit
Is your die fair—1 more
time.
Roll your die 120 times. Write down the
number of each roll.
Example:
Observed
25 20
35 20
19 20
14 20
Expected
16 20
11 20
Dice ---Requirements
 What are the requirements for the Chisquare test? EASI
 E—eighty percent of the expected cells
are greater than 5. (not observed cells—
expected cells!)
 A and
 S—representative sample.
 I—Independence.
Run test!!
 Note—Chi-square test is not normal.
 Must figure out the chi-square sum and
degrees of freedom.
 Chi-square sum: (O-E)^2/E and then add
them all up. Look up on formula sheets.
 On calculator X^2 cdf(x^2, 10^99, df)---also
use formula sheets.
 Sum is 1.25 + 11.25 + .05 + 1.8 + .8 + 4.05
 = 19.2
Run test!!
 X^2cdf(19.2, 10^99,5) = .0018
Reject Ho @ alpha .01. There is evidence to
suggest my die is not balanced.
What about Ho and Ha?
 There are no symbols for the chi-square.
However, it is always one-sided, even
though the word “different” is used.
 Ho: The dice will roll as you would expect.
(there is an implied = here)
 Ha: the die is different than you would
expect.
Demographics.
 Rancho is approximately 52% Hispanic, 27%
Asian, 16% white and 5% other.
 Does Mr. Pines’ AP stats classes reflect this
diversity? Run the appropriate test, verify
your requirements, and write a conclusion.
Baseball Bats
 There have been some major bat
changes for the 2011 season. Aluminum
baseball bats have been regulated so
that they meet certain safety standards.
After 5 games this season, coach Pines
has noticed significant reductions in
power numbers such as 2B’s, 3B’s, and
HR’s…..Of course he would like to test
his hypothesis.
Baseball Bats
 Run a Chi-Squared two-way table test to
see if there is an association between the
power numbers and types of bats.
 Also run a 2-Prop. Z Test between the
types of bats used.
 If these are done correctly, Z2 = X2
Hypotheses
 Ho: There is no association between type
of bats and extra base hits
 Ha: There is an association between type
of bats and extra base hits
Assumptions/Conditions
 E---All expected counts > 5
 S----We have a random sample of 23
schools hitting stats for the first 5 games
of the 2010 and 2011 baseball seasons
 I----We can assume that all stats are
independent of other teams stats
Observed and Expected
Counts
2010(BESR Bats)
2011(BBCOR Bats)
Singles
672
(695.26)
703
(679.74)
Extra Base Hits
313
(289.74)
260
(283.26)
 X2 = 5.35
 P-value = .0207
 This p-value is low enough to reject at
the 5% level.
 There is evidence to suggest that there
may be an association between the types
of bats and extra base hits
Gasoline
 A large distributor of gasoline claims that
60% of all cars stopping at their service
stations choose regular unleaded gas
and that premium and supreme are each
selected 20% of the time. To investigate
this claim, researchers collected data
from a random sample of drivers who put
gas in their vehicles at the stations in a
large city. Here are the results:
Gasoline Selected
Regular
Premium
Supreme
261
51
88
Carry out a significance test of the distributor’s
claim. Use a 5% significance level
Referrals vs Days of week
Monday
Tuesday
Wednesday
Thursday
Friday
12
5
9
4
15
The table shows the number of students
referred for disciplinary reasons to the
principals office, broken down by day of the
week.
Are referrals related to the day of the week?
Testing M&M’s
 The Mars company has always claimed
that the color distribution of their M&M’s
follow a certain proportion as follows:
Brown
Red
Yellow
Green
Orange
Blue
13%
13%
14%
16%
20%
24%
Check the M&M’s that were given to you and run a ChiSquare GOF test to see if their claim is accurate.
Do not eat your M&M’s until you have your observed and
expected counts completed!
Hypotheses
 Ho: My bag of M&M’s follow the same
color distribution as the Mars company
claim.
 Ha: My bag of M&M’s follows a different
color distribution as the Mars company
claim.
Assumptions/Conditions
 ___% of expected counts >5
 My bag of M&M’s can be considered an
independent random sample of M&M’s