106 Statistical Test of the Population Proportion

106.1 Theory

106.1.1 Statistical Hypothesis: Testing Population Proportion - Population

The Bernouilli Distribution (as described in Chapter 12) is used with

\[ \begin{cases}\text{Characteristic $A$ with probability P}(A) = \pi \\\text{Characteristic }\bar{A}\text{ with probability P}(\bar{A}) = 1 - \pi\end{cases} \]

106.1.2 Statistical Hypothesis: Testing Population Proportion - Sample

The sample statistic is defined as

\[ \begin{cases}p = \frac{\text{\# successes in the sample}}{\text{\# observations in the sample}} \\p = \text{sample proportion}\end{cases} \]

Repeated Bernoulli trials have a Binomial Distribution (cfr. Chapter 13). The Binomial Distribution, however, can be fairly well approximated by the Normal Distribution when the sample size is sufficiently large and the proportion is not too close to 0 or 1. For practical purposes, we therefore define the Sample Distribution as

\[ p \sim \text{N}\left( \pi, \frac{\pi (1-\pi)}{n} \right) \]

Hence, as a rule of thumb, if $n \pi \geq 10$ and $n (1-\pi) \geq 10$, we use

\[ u = \frac{p - \pi}{\sqrt{\frac{\pi (1-\pi)}{n}}} \sim \text{N}(0,1) \]

For moderate sample sizes, a continuity correction can improve the normal approximation (especially when the test statistic is close to the critical boundary). For very large samples the effect is usually small; the worked examples below omit the correction unless stated otherwise.

106.1.3 Summary

Table 106.1: Estimation of Population Proportion -- Hypothesis & Critical Regions

Null Hypothesis H$_0$	Alternative Hypothesis H$_A$	Critical Region
$\pi \leq \pi_0$	$\pi > \pi_0$	$p \geq \pi_0 + u_{\alpha} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}$
$\pi \geq \pi_0$	$\pi < \pi_0$	$p \leq \pi_0 - u_{\alpha} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}$
$\pi = \pi_0$	$\pi \neq \pi_0$	$\begin{cases} p \geq \pi_0 + u_{\frac{\alpha}{2}} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}\\ p \leq \pi_0 - u_{\frac{\alpha}{2}} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}} \end{cases}$

106.2 Examples

106.2.1 Statistical Hypothesis: Testing Population Proportion -- Example 1: Critical Value (Region)

106.2.1.1 Problem

Find the critical value if the following information is available:

Sample Size $n = 50$
# Successes = 17
Sample Proportion $p = 0.34$
Null Hypothesis H$_0: \pi_0 = 0.50$
Type I Error $\alpha = 0.05$

We treat this problem as a one-sided Hypothesis Test:

\[ \begin{cases}\text{H}_0: \pi \geq \pi_0 \\\text{H}_A: \pi < \pi_0\end{cases} \]

106.2.1.2 Solution

\[ \text{P} \left( \frac{p - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}} \leq \frac{c - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}} \right) = 0.05 \]

From Appendix E we know that

\[ \frac{c - \pi_0}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}} = -1.645 \]

which implies that

\[ \begin{align*}c &= \pi_0 - 1.645 \sqrt{\frac{\pi_0 (1-\pi_0)}{n}} \\c &= 0.50 - 1.645 \sqrt{\frac{0.50 \times 0.50}{50}} \\c &= 0.50 - (1.645 \times 0.0707) = 0.3837\end{align*} \]

106.2.1.3 Conclusion

Since $p = 0.34$ is smaller than $c = 0.3837$, we reject the Null Hypothesis H$_0: \pi_0 = 0.50$.

106.2.1.4 Software

The R module is available on the public website:

https://compute.wessa.net/rwasp_hypothesisprop1.wasp

To compute this Hypothesis Test on your local machine, the following script can be used in the R console:

library(Hmisc)
par1 = 50 #Sample size 
par2 = 0.34 #Sample Proportion
par3 = 0.50 #Null hypothesis 
par4 = 0.05 #Type I error (alpha) 
# Testing Population Proportion (normal approximation)
if (par2 < par3) {
  ucv <- qnorm(par4)
} else {
  ucv <- -qnorm(par4)
}
# 1-sided critical value
cv1 <- par3 + ucv * sqrt(par3 * (1-par3) / par1)
print(cv1)
cv2low <- par2 - abs(qnorm(par4/2)) * sqrt(par3 * (1-par3) / par1)
cv2upp <- par2 + abs(qnorm(par4/2)) * sqrt(par3 * (1-par3) / par1)
z21 <- qnorm(par4/2)^2 / par1
z2 <- qnorm(par4/2)^2 / (2*par1)
z24 <- qnorm(par4/2)^2 / (4*par1^2)
cv2lowexact <- (par2 + z2 - abs(qnorm(par4/2)) * 
sqrt(par3 * (1-par3) / par1 + z24)) / (1 + z21)
cv2uppexact <- (par2 + z2 + abs(qnorm(par4/2)) * 
sqrt(par3 * (1-par3) / par1 + z24)) / (1 + z21)
z11 <- qnorm(par4)^2 / par1
z1 <- qnorm(par4)^2 / (2*par1)
z14 <- qnorm(par4)^2 / (4*par1^2)
cv1lowexact <- (par2 + z1 - abs(qnorm(par4)) * 
sqrt(par3 * (1-par3) / par1 + z14)) / (1 + z11)
cv1uppexact <- (par2 + z1 + abs(qnorm(par4)) * 
sqrt(par3 * (1-par3) / par1 + z14)) / (1 + z11)
# 1-sided test
if (par2 < par3) {
  if (par2 < cv1) {
    print('Reject the Null Hypothesis')
  } else {
    print('Do not reject the Null Hypothesis')
  }
} else {
  if (par2 > cv1) {
    print('Reject the Null Hypothesis')
  } else {
    print('Do not reject the Null Hypothesis')
  }
}
# 2-sided Confidence Interval
dum <- paste('[',cv2low)
dum <- paste(dum,',')
dum <- paste(dum,cv2upp)
dum <- paste(dum,']')
print(dum)
# 2-sided test
if ((par3 < cv2low) | (par3 > cv2upp)) {
  print('Reject the Null Hypothesis')
} else {
  print('Do not reject the Null Hypothesis')
}
# Testing Population Proportion (Agresti-Coull method)
# Left 1-sided confidence interval
dum <- paste('[',cv1lowexact)
dum <- paste(dum,', 1 ]')
print(dum)
# Right 1-sided confidence interval
dum <- paste('[ 0 ,',cv1uppexact)
dum <- paste(dum,' ]')
print(dum)
# 2-sided Confidence Interval
dum <- paste('[',cv2lowexact)
dum <- paste(dum,',')
dum <- paste(dum,cv2uppexact)
dum <- paste(dum,']')
print(dum)
(re <- binconf(par2*par1,par1,par4,method='exact'))
(re1 <- binconf(par2*par1,par1,par4*2,method='exact'))
(rw <- binconf(par2*par1,par1,par4,method='wilson'))
(rw1 <- binconf(par2*par1,par1,par4*2,method='wilson'))
# Left 1-sided confidence interval
dum <- paste('[',re1[2])
dum <- paste(dum,', 1 ]')
print(dum)
# Right 1-sided confidence interval
dum <- paste('[ 0 ,',re1[3])
dum <- paste(dum,' ]')
print(dum)
# 2-sided Confidence Interval
dum <- paste('[',re[2])
dum <- paste(dum,',')
dum <- paste(dum,re[3])
dum <- paste(dum,']')
print(dum)
# Left 1-sided confidence interval
dum <- paste('[',rw1[2])
dum <- paste(dum,', 1 ]')
print(dum)
# Right 1-sided confidence interval
dum <- paste('[ 0 ,',rw1[3])
dum <- paste(dum,' ]')
print(dum)
# 2-sided Confidence Interval
dum <- paste('[',rw[2])
dum <- paste(dum,',')
dum <- paste(dum,rw[3])
dum <- paste(dum,']')
print(dum)

[1] 0.3836913
[1] "Reject the Null Hypothesis"
[1] "[ 0.201409617565032 , 0.478590382434968 ]"
[1] "Reject the Null Hypothesis"
[1] "[ 0.234929169412309 , 1 ]"
[1] "[ 0 , 0.461497450039298  ]"
[1] "[ 0.217860759029623 , 0.484970472693052 ]"
 PointEst     Lower     Upper
     0.34 0.2120547 0.4876525
 PointEst    Lower     Upper
     0.34 0.229548 0.4652991
 PointEst     Lower     Upper
     0.34 0.2243695 0.4784617
 PointEst     Lower     Upper
     0.34 0.2405721 0.4558545
[1] "[ 0.229547961591342 , 1 ]"
[1] "[ 0 , 0.465299081064598  ]"
[1] "[ 0.212054713435382 , 0.487652487721588 ]"
[1] "[ 0.240572121758134 , 1 ]"
[1] "[ 0 , 0.455854497693472  ]"
[1] "[ 0.224369492225001 , 0.478461739497674 ]"

106.2.2 Statistical Hypothesis: Testing Population Proportion -- Example 2: P-Value

106.2.2.1 Problem

Find the probability value if the following information is available:

Sample Size $n = 50$
# Successes = 17
Sample Proportion $p = 0.34$
Null Hypothesis H$_0: \pi_0 = 0.50$
Type I Error $\alpha = 0.05$

We treat this problem as a one-sided Hypothesis Test:

\[ \begin{cases}\text{H}_0: \pi \geq \pi_0 \\\text{H}_A: \pi < \pi_0\end{cases} \]

106.2.2.2 Solution

\[ u = \frac{p - \pi_0}{\sqrt{\frac{\pi_0 (1-\pi_0)}{n}}} = \frac{0.34 - 0.50}{\sqrt{\frac{0.50 \times 0.50}{50}}} = \frac{-0.16}{\sqrt{0.005}} = -2.2627 \]

for which the probability can be obtained

\[ \text{P}(u < -2.2627) = 0.0118 \]

106.2.2.3 Conclusion

Since $\text{P}(u < -2.2627) = 0.0118$ is smaller than $\alpha = 0.05$, we reject the Null Hypothesis H$_0: \pi_0 \geq 0.50$

106.2.2.4 Software

The R module is available on the public website:

https://compute.wessa.net/rwasp_hypothesisprop2.wasp

To compute this Hypothesis Test on your local machine, the following script can be used in the R console:

par1 = 50 #Sample size 
par2 = 0.34 #Sample Proportion
par3 = 0.50 #Null hypothesis 
par4 = 0.05 #Type I error (alpha) 
u <- (par2 - par3) / sqrt(par3 * (1-par3) / par1)
(pu <- pnorm(-abs(u)))

[1] 0.01182581

Null Hypothesis H\(_0\)	Alternative Hypothesis H\(_A\)	Critical Region
\(\pi \leq \pi_0\)	\(\pi > \pi_0\)	\(p \geq \pi_0 + u_{\alpha} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}\)
\(\pi \geq \pi_0\)	\(\pi < \pi_0\)	\(p \leq \pi_0 - u_{\alpha} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}\)
\(\pi = \pi_0\)	\(\pi \neq \pi_0\)	\(\begin{cases} p \geq \pi_0 + u_{\frac{\alpha}{2}} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}\\ p \leq \pi_0 - u_{\frac{\alpha}{2}} \sqrt{\frac{\pi_0 (1 - \pi_0)}{n}} \end{cases}\)