Chapter 12
Statistical Inference: Other OneSample Test Statistics
I One-Sample z Test for a Population
Proportion, p
A. Introduction to z Test for a Population
Proportion
1
1. The binomial function rule
p( X  r)  n Cr pr ( p  1)nr
can be used to determine the probability of r
successes in n independent trials.
2. When n is large, the normal distribution can be
used to approximate the probability of r or more
successes. The approximation is excellent if
(a) the population is at least 10 times larger than
the sample and (b) np0 > 15 and n(1 – p0) > 15,
where p0 is the hypothesized proportion.
2
B. z Test Statistic for a Proportion
1.
z
pˆ  p0
p0 1  p0  n
p̂ = sample estimator of the population
proportion
number of successes in the random sample

number of observations in the random sample
p0 = hypothesized population proportion
n = size of the sample used to compute p̂
3
2.
ˆ p  p0 1  p0  n is an estimator of the
population standard error of a proportion,
p 
p(1  p / n,
where p denotes the population proportion.
C. Statistical Hypotheses for a Proportion
H0 : p  p0
H0 : p  p0
H0 : p  p0
H1 : p  p0
H1 : p  p0
H1 : p  p0
4
D. Computational Example
1. Student Congress believes that the proportion of
parking tickets issued by the campus police this
year is greater than last year. Last year the
proportion was p0 = .21.
2. To test the hypotheses H0 : p  .21
H1 : p  .21
they obtained a random sample of n = 200
students and found that the proportion who
received tickets this year was pˆ  .27.
5
z
pˆ  p0

p0 1  p0 
n
.27  .21
 2.08
.211  .21
200
z.05 = 1.645
3. The null hypothesis can be rejected; the campus
police are issuing more tickets this year.
6
E. Assumptions of the z Test for a Population
Proportion
1. Random sampling from the population
2. Binomial population
3. np0 > 15 and n(1 – p0) > 15
4. The population is at least 10 times larger than the
sample
7
II One-Sample Confidence Interval for a
Population Proportion, p
A. Two-Sided Confidence Interval
1. p̂  z /2
2. ˆ p 
p̂(1  p̂)
p̂(1  p̂)
 p  p̂  z /2
n
n
pˆ 1  pˆ  n is an estimator of the
population standard error of a proportion.
8
B. One-Sided Confidence Interval
1. Lower confidence interval
p̂  z
p̂(1  p̂)
p
n
2. Upper confidence interval
p  p̂  z
p̂(1  p̂)
n
9
C. Computational Example Using the Parking
Ticket Data
1. Two-sided 100(1 – .05)% = 95% confidence
interval
p̂  z /2
p̂(1  p̂)
p̂(1  p̂)
 p  p̂  z /2
n
n
.27(1  .27)
.27(1  .27)
.27  1.96
 p  .27  1.96
200
200
.208  p  .332
10
2. One-sided 100(1 – .05)% = 95% confidence
interval
p̂  z
p̂(1  p̂)
p
n
.27(1  .27)
.27  1.645
p
200
.218  p
11
3. Comparison of the one- and two-sided confidence
intervals
Two-sided interval
L1 = .208
.20
L 2 = .332
.25
.30
.35
p
One-sided interval
L = .218
1
.20
.25
p
.30
.35
12
D. Assumptions of the Confidence Interval for a
Population Proportion
1. Random sampling from the population
2. Binomial population
3. np0 > 15 and n(1 – p0) > 15
4. The population is at least 10 times larger than the
sample
13
III Selecting a Sample Size, n
A. Information needed to specify n
1. Acceptable margin of error, m*, in
estimating p. m* is usually between .02
and .04.
2. Acceptable confidence level: usually .95
for z.05 or z.05/2
3. Educated guess, denoted by p*, of the
likely value of p
14
B. Computational Example for the Traffic
Ticket Data
1. One-sided confidence interval, let m* = .04,
z.05 = 1.645, and p* = .27
2
 z.05 
n
p * (1 p*)

 m *
2
 1.645 
n
.27(1  .27)  333

 .04 
15
C. Conservative Estimate of the Required Sample
Size
1. If a researcher is unable to provide an educated
guess for m*, a conservative estimate of n is
obtained by letting p* = .50.
2
 z.05 
n
p * (1 p*)

 m *
2
 1.645 
n
.50(1  .50)  423

 .04 
16
IV One-Sample t Test for Pearson’s Population
Correlation
A. t Test for 0 = 0 (Population Correlation
Is Equal to Zero)
1. Values of | r | that lead to rejecting one of the
following null hypotheses are obtained from
Appendix Table D.6.
H0 :   0
H0 :   0
H0 :   0
H1 :   0
H1 :   0
H1 :   0
17
Appendix Table D.6. Critical Values of the Pearson r
Degrees of
Level of Significance for a One-Tailed Test
.005
.05
.025
.01
Freedom
  n2
Level of Significance for a Two-Tailed Test
.05
.10
.02
.01
8
0.549
0.632
0.716
0.765
10
0.497
0.576
0.658
0.708
20
0.360
0.423
0.492
0.537
30
0.296
0.349
0.409
0.449
60
0.211
0.250
0.274
0.325
100
0.164
0.195
0.230
0.254
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1. Table D.6 is based on the t distribution and t
statistic
t
r n2
1 r 2
with n  2 degrees of freedom
B. Computational Example Using the Girl’s
Basketball Team Data (Chapter 5)
1. H0 :   0, r = .84, n = 10, and r.05, 8 = .549
2. r.05, 8 = .549 is the one-tailed critical value from
Appendix Table D.6.
19
1. Because r = .84 > r.05, 8 = .549, reject the null
hypothesis and conclude that player’s height and
weight are positively correlated.
20
C. Assumptions of the t Test for Pearson’s
Population Correlation Coefficient
1. Random sampling
2. Population distributions of X and Y are
approximately normal.
3. The relationship between X and Y is linear.
21
4. The distribution of Y for any value of X is
normal with variance that does not depend on the
X value selected and vice versa.
V One-Sample Confidence Interval for
Pearson’s Population Correlation
A. Fisher’s r to Z Transformation
1. r is bounded by –1 and +1; Fisher’s Z can
exceed –1 and +1.
22
Appendix Table D.7 Transformation of r to Z
r
Z
r
Z
r
Z
r
Z
0.200 0.203
0.400 0.424
0.600 0.693
0.800 1.099
0.225 0.229
0.425 0.454
0.625 0.733
0.825 1.172
0.250 0.255
0.450 0.485
0.650 0.775
0.850 1.256
0.275 0.282
0.475 0.517
0.675 0.820
0.875 1.354
0.300 0.310
0.500 0.549
0.700 0.867
0.900 1.472
0.325 0.337
0.525 0.583
0.725 0.918
0.925 1.623
0.350 0.365
0.550 0.618
0.750 0.973
0.950 1.832
0.375 0.394
0.575 0.655
0.775 1.033
0.975 2.185
23
B. Two Sided Confidence Interval for
 Using
Fisher’s Z Transformation
1. Begin by transforming r to Z. Then obtain
a confidence interval for ZPop
Z   z.05/ 2
1
1
 Z Pop
  Z   z.05/ 2
n3
n3
2. A confidence interval for r is obtained by
transforming the lower and upper confidence
limits for ZPop into r using Appendix Table D.6 .
24
C. One-Sided Confidence Interval for 
1. Lower confidence limit
Z   z.05
1
 Z Pop

n3
2. Upper confidence limit
Z Pop
  Z   z.05
1
n3
25
D. Computational Example Using the Girl’s
Basketball Team Data (Chapter 5)
1. r = .84, n = 10, and Z = 1.221
Z   z.05
1
 Z Pop

n3
1
1.221  1.645
 Z Pop

10  3
.599  Z Pop

.54  
26
2. Graph of the confidence interval for 
L
.50
1 = .54
.55

.60
.65
3. A confidence interval can be used to test
hypotheses for any hypothesized value of 0.
For example, any hypothesis for which 0 ≤ .54
could be rejected.
27
E. Assumptions of the Confidence Interval for
Pearson’s Correlation Coefficient
1. Random sampling
2.  is not too close to 1 or –1
3. Population distributions of X and Y are
approximately normal
4. The relationship between X and Y is linear
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5. The distribution of Y for any value of X is
normal with variance that does not depend on the
X value selected and vice versa.
VI Practical Significance of Pearson’s
Correlation
A. Cohen’s Guidelines for Effect Size
 r = .10 is a small strength of association
 r = .30 is a medium strength of association
 r = .50 is a large strength of association
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