1. STATISTICS &
PROBABILITY
Z-test on Proportion
Chi-square test on K-proportions

2. Z-test on Proportion & Chi-
square test on K-proportions
Test of hypotheses concerning proportions are
important in many areas.
Some of which are as follows:
• Suppose the production manager wants to know if there is
evidence of improvement in the production processes
by testing if the proportion or the number of defectives
had been reduced.

3. Z-test on Proportion & Chi-
square test on K-proportions
• Hotel owners are interested in knowing if there is a
significant difference between the proportion of
clients who are satisfied and not satisfied in the
services provided by the hotel personnel.
• A candidate would want to know her/his chances of
winning by finding out if the proportion of people who
will vote for her/him is significantly higher than those
who are most likely to vote for her/his opponents.

4. Z-test on Proportion & Chi-
square test on K-proportions
• The discipline officer wants to know if the number new
policy on smoking has reduced the number of smokers
in CSB.
• The guidance counselor wants to know if the number of
female dress code violators is significantly higher than
the number of male dress code violator.

5. Z-test on Proportion & Chi-
square test on K-proportions

6. Z-test on Proportion
For testing the significance of difference between
sample proportion and hypothesized value
x − npo
Z= x = the number of successes
npo (1 − po ) n= the number of sample
Po = the hypothesized proportion
Using PHStat: Go to…
“One-Sample Tests; Z-Test on Proportions…”

7. Z-test on Proportion
For testing the significance ofGo to…
Using PHStat: difference between
“Two-Sample Tests; Z-Test for Differences in Two Prop…”
two sample proportions
Or Chi-Square Test for Differences in Two Prop…
x1 x2
− x1 + x2
n1 n2 and p =
Z=
1 1  n1 + n2
p (1 − p ) + 
n 
 1 n2 
x1 = the number of successes in the first group
x2 = the number of successes in the second group
n1 = the number of sample in the first group
n2 = the number of sample in the second group

8. TESTING A CLAIM ABOUT STANDARD
DEVIATION OR VARIANCE
Test Statistics for Testing Hypothesis About σ or σ2 :
Chi-square distribution:
x 2
=
( n −1) s 2
σ 2
where: n = sample size
s2 = sample variance
σ2 = population variance (given in the null
hypothesis

9. Criterion:
1.One − tailed left directional
2 2
Reject H 0 if, x ≤ x 1−α
2.One − tailed right directional
2 2
Reject H 0 if, x ≥ x α
3.Two − tailed
2 2 α 2 2α
Reject H 0 if, x ≤ x 1− and x ≥x
2 2

10. Chi-square test on two or
K-Proportions
X =∑
2 ( o − e) 2
e
where: “o” stands for observed frequencies
and “e” stands for the expected frequencies.
Using PHStat: Go to…
“Multiple-Sample Tests; Chi-Square Test”

11. EXERCISES: Test Concerning Proportions
1. At a certain college, it is estimated that 25%of
the students have cars on campus. Does this
seem to be valid estimate if, in a random
sample of 90 college students, 28 are found to
have cars. Use a 0.05 level of confidence.
2. A cigarette-manufacturing firm distributes two
brands of cigarettes. It is found that 56 of 200
smokers prefer brand A and that 29 of 150
smokers prefer brand B, can we conclude at
0.01 level of significance that brand A outsells
brand B?

12. 3. It has been claimed that 55% of students dislike
mathematics. When a survey as conducted, it
showed that 153 of 600 students dislike
mathematics. Test if the claim is too high at
α = 0.05 .
4. In a factory of baby dresses, one production
process yielded 30 defective pieces in a random
sample of 400, while another yielded 17
defective pieces in a random sample of 300. Is
there a significant difference between the
proportions of defective baby dresses? Test at
0.01 level of significance.

13. 5. The Office of the Dean will conduct a research
on the study grants recipients in a certain
school. It was reported that 25% of the present
grantees are ineligible and should not have
been receiving any grants. You are hired to
investigate the claim and in a survey you
conducted, you found that out of 200 grantees,
30 should have been disqualified. At α = 0.05
level, should the Dean’s Office claim be
rejected?

14. 6. With individual lines at its various windows, the
CSB bank found that the standard deviation for
normally distributed waiting lines on Friday
afternoons was 6.2 minutes. The bank
experimented with a single main waiting line and
found that for that for random sample of 45
customers, the waiting times have a standard
deviation of 3.8 mins. Based on previous
studies, we can assume that the waiting lines
are normally distributed. At α = 0.05, test the
claim that a single line causes lower variation
among the lines.

15. 7. In a study of the wide ranges in the
academic success of college freshmen,
one various factor is the amount spend in
studying. At the 0.01 level of significance,
test the claim that the standard deviation
is more than 4 hours. The sample consists
of 70 randomly selected freshmen who
have a standard deviation of 5.33 hours.

16. 8. ABS Corporation have been successfully
manufacturing electronic parts with a
standard deviation of 43.7 from the
existing line. After the installation of the
new line a sample of 50 products was
inspected from the line and found that the
standard deviation was 54.7. Has the
standard deviation of the total products
changed with the new equipment? Use
α = 0.05.