1. Chapter 10 – Estimation and Testing for Population Proportions
Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 10 – Estimation and Testing for Population Proportions
Slides prepared by Jeff Heyl, Lincoln University
©2003 South-Western/Thomson Learning™

2. Point Estimate for a Population Proportion
where n = sample size and
x = the observed number of successes in the sample
p = estimate of p
= proportion of sample having a specified attribute = ^ x n

3. Confidence Interval for a Population Proportion
Table A.8 provides 90% and 95% confidence interval values for small samples
From A.8 a 95 % confidence interval pL to pU = .519 to .957
For n = 15 and x = 12 12 15
p = ^
Example 10.1

4. Confidence Interval for a Population Proportion
p(1 - p)
n - 1 ^
sp =
Estimated standard error
(1-) 100% confidence interval for large samples
p - Z/ to p + Z/2 ^
p(1 - p)
n - 1

5. Choosing the Sample Size
With 95% confidence
E = 1.96
p(1 - p)
n - 1 ^
With E = .02
E = .02 = 1.96
(.0867)(.9133)
n - 1
n = 761.5
n =
Z/2p(1 - p) E2 ^ 2

6. Curve of Values | .5 1 p ^
p(1 - p)
.25 –
Figure 10.1

7. Hypothesis Testing Using a Small Sample
Ho: p = po
Ha: p ≠ po
Two-Tailed Test
Obtain (1- ) • 100% confidence interval
Reject Ho if po does not lie between pL and pU
Fail to reject Ho if pL ≤ po ≤ pU

8. Hypothesis Testing Using a Small Sample
Ho: p ≤ po
Ha: p > po
One-Tailed Test
Obtain (1- 2) • 100% confidence interval
Reject Ho if po < pL
Fail to reject Ho if po ≥ pL

9. Hypothesis Testing Using a Small Sample
Ho: p ≥ po
Ha: p < po
One-Tailed Test
Obtain (1- 2) • 100% confidence interval
Reject Ho if po > pL
Fail to reject Ho if po ≤ pU

10. Hypothesis Testing Using a Large Sample
Ho: p = po
Ha: p ≠ po
reject Ho if |Z| > Z/2
Two-Tailed Test
where
Z =
p - po
po(1 - po) n ^

11. Hypothesis Testing Using a Large Sample
Ho: p ≤ po
Ha: p > po
reject Ho if Z > Z
One-Tailed Test
Ho: p ≥ po
Ha: p < po
reject Ho if Z < -Z
Z =
(point estimate) - (hypothesized value)
(stanard deviation of point estimator)

12. Z Curve Showing p-Value
p-value = area =
= .008 Z
Figure 10.2

13. Z Curve Showing p-Value
p-value = 2(area)
= 2( )
= 2(.0018)
= .0036 Z
Figure 10.3

14. Excel Z Test
Figure 10.4

15. Comparing Two Population Proportions (Large Independent Samples)
Standard error
sp -p = 1 2 ^
p1(1 - p1)
n1 - 1
p2(1 - p2)
n2 - 1
Confidence interval
(p1 - p2) - Z/2 ^
p1(1 - p1)
n1 - 1
p2(1 - p2)
n2 - 1 +
to (p1 - p2) + Z/2

16. Sample Size -Two Populations
where
E = Z/
p1(1 - p1)
n1 - 1 ^
p2(1 - p2)
n2 - 1
n1 =
Z/2(A + B) E2 2
n2 =
Z/2(C + B)
A = p1(1 - p1)
B = p1p2(1 - p1)(1 - p2)
C = p2(1 - p2)

17. Hypothesis Test for Population Proportion
Two-Tailed Test
Ho: p1 = p2
Ha: p1 ≠ p2
reject Ho if |Z| > Z/2
For test statistic
Z =
p1(1 - p1)
n1 - 1 ^
p2(1 - p2)
n2 - 1 +
p1 - p2

18. Hypothesis Test for Population Proportion
One-Tailed Test
Ho: p1 ≤ p2
Ha: p1 > p2
reject Ho if |Z| > Z
Ho: p1 ≥ p2
Ha: p1 < p2
reject Ho if |Z| < -Z
For test statistic
Z =
p1(1 - p1)
n1 - 1 ^
p2(1 - p2)
n2 - 1 +
p1 - p2

19. Z Curve Showing p-Value
p-value = 2(area)
= 2( )
= .4966 Z
Figure 10.5

20. Z Curve Showing p-Value
p-value = area =
= .0005 Z
Figure 10.6

21. Excel Solution - Z Test
Figure 10.6