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Researchers designed a survey to compare the proportions of children
who come to school without eating breakfast in two low-income
elementary schools. An SRS of 80 students from School 1 found that 19
had not eaten breakfast, At School 2, an SRS of 150 students included
26 who had not had breakfast. More than 1500 students attend each
school. Do these data give convincing evidence of a difference in the
population proportions? Carry out a significance test at the a = 0.05 to
support your answer.
Testing for a Difference
in Proportions, p1 – p2
Suppose we have two independent binomial experiments.
We would like to test if the two population proportions are equal.
Binomial Experiment 1
Binomial Experiment 2
n1 = number of trials
n2 = number of trials
x1 = number of successes
x2 = number of successes
p1 = population probability of
success on a single trial
p2 = population probability of
success on a single trial
Testing for a Difference in Proportions
For large values of n1 and n2
r1 r2
p1  p 2  
n1 n 2
Is closely approximated by a normal distribution with
  p1  p2

p1q1 p2 q 2

n1
n2
where q 1  1  p1 and q 2  1  p2
Testing for a Difference in Proportions
The test statistic is as follows:
z
p1  p2

pc 1  p c
n1
  p 1  p 
x1  x 2
where pc 
n1  n 2
c
c
n2
How to test a difference of proportions p1 – p2
Consider two independent binomial experiments
Binomial Experiment 1
Binomial Experiment 2
n1 = number of trials
n2 = number of trials
x1 = number of successes out of
n1 trials
x2 = number of successes out of
n2 trials
x1
p1 
n1
x2
p2 
n2
p1 = population probability of
success on a single trial
p2 = population probability of
success on a single trial
The Test Procedure
1. Use the null hypothesis of no difference, H0: r1 = r2. Choose
the alternate hypothesis in the context of the problem. Set the
level of significance a.
2. The null hypothesis is of no difference r1 = r2; therefore
pooled (ie., combined) best estimates for the population
probabilities of success and failure are
x1  x 2
pc 
and q c  1  pc
n1  n 2
The Test Procedure
The number of trials should be sufficiently large so that all four
quantities n1pc , n1qc , n2pc , n2qc are each larger than 10.
Compute the sample test statistic
z
p1  p2
pc qc pc qc

n1
n2
The Test Procedure
3. Use the standard normal distribution and type of test,
one-tailed or two-tailed, to find the P-value corresponding
to the test statistic.
4. Conclude the test. If P-value < a, then reject H0. If Pvalue > a, then fail to reject the null.
5. State your conclusion in the context of the problem.
Researchers designed a survey to compare the proportions of children
who come to school without eating breakfast in two low-income
elementary schools. An SRS of 80 students from School 1 found that 19
had not eaten breakfast, At School 2, an SRS of 150 students included
26 who had not had breakfast. More than 1500 students attend each
school. Do these data give convincing evidence of a difference in the
population proportions? Carry out a significance test at the a = 0.05 to
support your answer.