Hypothesis Testing
on
Sample Proportion
One-sample and
Two-Sample
Proportions
Test about the Population
Proportion - “p”
Test-statistic:
z=
^ - p0
p
___________
√
p0 (1-p0)
n
Ho: p = p0 (compute the z-statistic)
Example:
LeRoy, a starting player for a major college basketball
team, made only 40% of his free throws last season.
During the summer, he worked on developing a softer
shot in hopes of improving his free throw accuracy. In
the first eight games of this season, LeRoy made 25 free
throws in 40 attempts. You want to investigate whether
LeRoy’s work over the summer will result in a higher
proportion of free-throw successes this season. What
conclusion would you draw at the 1% significance level
about LeRoy’s free throw shooting?
Po =
.40
P-hat = 25/40
n=
40
Ho: Leroy’s proportion of free throw hits is
p=.40
Ho: Leroy’s proportion of free throw hits is
p>.40
SRS: we must be willing
to treat that his free throws are
random
Normality: Rule of Thumb #2
np≥10 40(.40)≥10 16≥10
nq≥10 40(.60)≥10 24≥10
Independence: One
Leroy
’
s
free
throw
is
Independent
sample z-test for
proportion
Z = 2.90
P-val = .0019
With a p-val = .0019 we have enough evidence to reject
the null hypothesis making the test significant at 1%
significance level. Therefore Leroy has improved his
Hypothesis Testing
on
Sample Proportion
One-sample and
Two-Sample
Proportions
Warm up
Any Miami Heat fan or arch-rival knows the team's very
large “SHAQilles heel”—the free-throw shooting of the
NBA's most dominant center, Shaquille O'Neal. Over
his NBA career, Shaq has made 53.3% of his free throws.
Shaquille O'Neal worked in the off-season with an
assistant coach on his free-throw technique. During the
first two games of the next season, Shaq made 26 out of
39 free throws.
(a) Do these results provide evidence that Shaq has significantly
improved his free-throw shooting?
(b) Describe a Type I error and a Type II error in this situation.
P = proportion of Shaq’s successful free throw hits
H0: p = 0.533
vs.
Ha: p > 0.533.
z = 1.67, P–value = 0.0475.
Therefore: we reject Ho and conclude that
Shaq improved his free throw shot training with
his coach. This makes our HT significant.
Type I error: concluding that Shaq has improved
his free–throwing when in fact he has not.
Type II error: concluding that Shaq has not
improved his free–throwing when in fact he has.
Remaining topics we need to cover:
1. 2-sample z-statistic
2. 2-sample t-statistic
3. Confidence interval for 2 samples
4. Comparing proportions of 2 populations
5. Significance test for combined sample proportions
6. Confidence interval for 2 proportions
7. Chi-Square