# Preliminaries

You use proptestGC() for inferential procedures regarding:

• one population proportion $$p$$;
• the difference of two population proportions, $$p_1 - p_2$$.

The function comes from the tigerstats package, so make sure that tigerstats is loaded:

require(tigerstats)

# One Proportion, From Data Frame (Read This!!)

In inference procedures regarding one proportion, binomtestGC() is preferred over proptestGC(), especially at smaller sample sizes. Nevertheless, read this section carefully. It talks about:

• what to count as a “success”,
• levels of confidence,
• types of Alternative Hypothesis,
• graphs of the P-value, and
• an option to limit output to the console

in ways that apply to all uses of proptestGC().

## Confidence Interval Only

In the m111survey data from the tigerstats package, suppose you want a 95%-confidence interval for:

$$p =$$ the proportion of all Georgetown College students who are female.

Use the function:

proptestGC(~sex,data=m111survey,
success="female")
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.448016            0.678745

Note that the success argument is set to “female” in order to make the function recognize $$p$$ as the population proportion of females, rather than the proportion of males (or of some other value of the variable in question).

To get any other level of confidence, you need to use the conf.level argument, indicating your desired level of confidence in decimal (not percentage) form.

For example, for a 90%-confidence interval for $$\mu$$, use

proptestGC(~sex,data=m111survey,
success="female",conf.level=0.90)
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 90% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.466564            0.660197

## Interval and Test

If you want a test of significance as well as the confidence interval, then

• use the p argument to set what the Null Hypothesis thinks that the value of $$p$$ is;
• use the alternative argument to specify the Alternative Hypothesis.

For example, if the hypotheses are:

$$H_0: p = 0.50$$

$$H_a: p > 0.50$$

Then use:

proptestGC(~sex,data=m111survey,p=0.50,
success="female",alternative="greater")
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.466564            1.000000
##
## Test of Significance:
##
##  H_0:  p = 0.5
##  H_a:  p > 0.5
##
##  Test Statistic:     z = 0.9571
##  P-value:        P = 0.1692

If the hypotheses are:

$$H_0: p = 0.50$$

$$H_a: p < 0.50$$

Then use:

proptestGC(~sex,data=m111survey,p=0.50,
success="female",alternative="less")
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.000000            0.660197
##
## Test of Significance:
##
##  H_0:  p = 0.5
##  H_a:  p < 0.5
##
##  Test Statistic:     z = 1.196
##  P-value:        P = 0.8842

If the hypotheses are:

$$H_0: p = 0.50$$

$$H_a: p \neq 0.50$$

Then use:

proptestGC(~sex,data=m111survey,p=0.50,
success="female",alternative="two.sided")
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.448016            0.678745
##
## Test of Significance:
##
##  H_0:  p = 0.5
##  H_a:  p != 0.5
##
##  Test Statistic:     z = 0.9571
##  P-value:        P = 0.3385

But note that the default value of alternative is “two.sided”, so if you want you could just leave it out and still get a two-sided test:

proptestGC(~sex,data=m111survey,p=0.50,
success="female")

## Graph of the P-Value

Anytime you want, you can get a graph of the $$P$$-value for your test, simply by setting the argument graph to TRUE:

proptestGC(~sex,data=m111survey,p=0.50,
success="female",graph=TRUE)
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  female  n estimated.prop
##      40 71         0.5634
##
##
## Inferential Results:
##
## Estimate of p:    0.5634
## SE(p.hat):    0.05886
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.448016            0.678745
##
## Test of Significance:
##
##  H_0:  p = 0.5
##  H_a:  p != 0.5
##
##  Test Statistic:     z = 0.9571
##  P-value:        P = 0.3385

## Limiting Output

Sometimes you don’t need R to print so much information to the console. If you want only the basics (such as a confidence interval, the test statistic and $$P$$-value), then set the verbose argument to FALSE:

proptestGC(~sex,data=m111survey,p=0.50,
success="female",verbose=FALSE)
##
##
## Inferential Procedures for a Single Proportion p:
##  Variable under study is sex
##  Continuity Correction Applied to Test Statistic
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.448016            0.678745
##
##  Test Statistic:     z = 0.9571
##  P-value:        P = 0.3385

# One Proportion, From Summary Data

Again note that binomtestGC() is preferred in this situation, especially at smaller sample sizes.

## Confidence Interval Only

Say that you have taken a simple random sample of size $$n=2500$$ from the population of all registered voters in the U.S., and you find that 1325 of them favor the Affordable Care Act. Suppose that you would like to make a confidence interval for

$$p =$$ the proportion of all registered voters in the U.S. who support the Act.

You don’t have the raw data present in a data frame, but you have enough summary data to use proptestGC(). You just have to set some new arguments:

• set x to 1325, the number of “successes” you are counting up;
• set n to 2500, the sample size.

So if you only want a 95%-confidence interval for $$p$$. use:

proptestGC(x=1325,n=2500)
##
##
## Inferential Procedures for a Single Proportion p:
##  Results based on Summary Data
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  successes    n estimated.prop
##       1325 2500           0.53
##
##
## Inferential Results:
##
## Estimate of p:    0.53
## SE(p.hat):    0.009982
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.510436            0.549564

## Interval and Test

If you also want to do a test of significance, again specify p and alternative. For example, to test the hypotheses

$$H_0: p = 0.50$$

$$H_a: p < 0.50$$

use:

proptestGC(x=1325,n=2500,
p=0.50,,alternative="less")
##
##
## Inferential Procedures for a Single Proportion p:
##  Results based on Summary Data
##  Continuity Correction Applied to Test Statistic
##
##
## Descriptive Results:
##
##  successes    n estimated.prop
##       1325 2500           0.53
##
##
## Inferential Results:
##
## Estimate of p:    0.53
## SE(p.hat):    0.009982
##
## 95% Confidence Interval for p:
##
##           lower.bound         upper.bound
##           0.000000            0.546419
##
## Test of Significance:
##
##  H_0:  p = 0.5
##  H_a:  p < 0.5
##
##  Test Statistic:     z = 3.025
##  P-value:        P = 0.9988

# Difference of Two Proportions, Data Frame

## Confidence Interval Only

Suppose

$$p_1 =$$ proportion of all GC females who believe they get enough sleep

$$p_2 =$$ proportion of all GC males who believe they get enough sleep

If you desire, say, an 85%-confidence interval for $$p_1 - p_2$$, then use:

proptestGC(~sex+enough_Sleep,data=m111survey,
success="yes",conf.level=0.85)
##
##
## Inferential Procedures for the Difference of Two Proportions p1-p2:
##   enough_Sleep grouped by sex
##
##
## Descriptive Results:
##
##        yes  n estimated.prop
## female  14 40         0.3500
## male    11 31         0.3548
##
##
## Inferential Results:
##
## Estimate of p1-p2:    -0.004839
## SE(p1.hat - p2.hat):  0.1143
##
## 85% Confidence Interval for p1-p2:
##
##           lower.bound         upper.bound
##           -0.169426           0.159749

## Interval and Test

If you want a 95%-confidence interval for $$p\_1 - p_2$$ and you would like to test the hypotheses:

$$H_0: p_1 - p_2 = 0$$

$$H_a: p_1 - p_2 \neq 0$$

then use:

proptestGC(~sex+enough_Sleep,data=m111survey,
success="yes",p=0)
##
##
## Inferential Procedures for the Difference of Two Proportions p1-p2:
##   enough_Sleep grouped by sex
##
##
## Descriptive Results:
##
##        yes  n estimated.prop
## female  14 40         0.3500
## male    11 31         0.3548
##
##
## Inferential Results:
##
## Estimate of p1-p2:    -0.004839
## SE(p1.hat - p2.hat):  0.1143
##
## 95% Confidence Interval for p1-p2:
##
##           lower.bound         upper.bound
##           -0.228929           0.219252
##
## Test of Significance:
##
##  H_0:  p1-p2 = 0
##  H_a:  p1-p2 != 0
##
##  Test Statistic:     z = -0.04232
##  P-value:        P = 0.9662

Notice that this time:

• you specified the Null’s belief about the value of $$p\_1-p_2$$ using the p argument;
• you did not have to set conf.level to 0.95 (the default value of the argument is 0.95 already);
• you did not have to specify the two-sided alternative (the default value of alternative is already “two.sided”).

## Order of the Groups

Suppose that in the previous situation you had defined:

$$p_1 =$$ proportion of all GC males who believe they get enough sleep

$$p_2 =$$ proportion of all GC females who believe they get enough sleep

Then for you, the first population is all GC males and the second population is all GC females. In order to guarantee that R abides by your choice, use the argument first:

proptestGC(~sex+enough_Sleep,data=m111survey,
success="yes",p=0,first="male")
##
##
## Inferential Procedures for the Difference of Two Proportions p1-p2:
##   enough_Sleep grouped by sex
##
##
## Descriptive Results:
##
##        yes  n estimated.prop
## male    11 31         0.3548
## female  14 40         0.3500
##
##
## Inferential Results:
##
## Estimate of p1-p2:    0.004839
## SE(p1.hat - p2.hat):  0.1143
##
## 95% Confidence Interval for p1-p2:
##
##           lower.bound         upper.bound
##           -0.219252           0.228929
##
## Test of Significance:
##
##  H_0:  p1-p2 = 0
##  H_a:  p1-p2 != 0
##
##  Test Statistic:     z = 0.04232
##  P-value:        P = 0.9662

# Difference of Two Proportions, Summary Data

## Confidence Interval Only

Suppose that you have taken two independent samples from two populations (or performed a completely randomized experiment with two treatment groups), and you have the following summary data:

Group Success Count Sample Size
group one 1325 2500
group two 905 1800

You need to provide the summary data to the arguments x and n as lists using the c() function. In each list, data from the first group should come first.

For a 95%-confidence interval for $$p_1 - p_2$$, use:

proptestGC(x=c(1325,905),n=c(2500,1800))
##
##
## Inferential Procedures for the Difference of Two Proportions p1-p2:
##  Results taken from summary data.
##
##
## Descriptive Results:
##
##         successes    n estimated.prop
## Group 1      1325 2500         0.5300
## Group 2       905 1800         0.5028
##
##
## Inferential Results:
##
## Estimate of p1-p2:    0.02722
## SE(p1.hat - p2.hat):  0.01544
##
## 95% Confidence Interval for p1-p2:
##
##           lower.bound         upper.bound
##           -0.003048           0.057492

## Interval and Test

Suppose that you want a 90%-confidence interval for $$p_1 - p_2$$ and that you would like to test the hypotheses:

$$H_0: p_1 - p_2 = 0$$

$$H_a: p_1 - p_2 > 0$$

Suppose also that you want a graph of the $$P$$-value. Then use:

proptestGC(x=c(1325,905),n=c(2500,1800),
p=0,conf.level=0.90,graph=TRUE)
##
##
## Inferential Procedures for the Difference of Two Proportions p1-p2:
##  Results taken from summary data.
##
##
## Descriptive Results:
##
##         successes    n estimated.prop
## Group 1      1325 2500         0.5300
## Group 2       905 1800         0.5028
##
##
## Inferential Results:
##
## Estimate of p1-p2:    0.02722
## SE(p1.hat - p2.hat):  0.01544
##
## 90% Confidence Interval for p1-p2:
##
##           lower.bound         upper.bound
##           0.001819            0.052626
##
## Test of Significance:
##
##  H_0:  p1-p2 = 0
##  H_a:  p1-p2 != 0
##
##  Test Statistic:     z = 1.763
##  P-value:        P = 0.07797