Confidence Interval for Proportions
As usual, the problems should be composed of the three components; THINK, SHOW and TELL.
THINK: The think in a confidence interval for proportions is to check and prove that 4 conditions
have been met. These conditions have to be met for you to be allowed to create a confidence interval.
1. Random Condition – it was a simple random sample drawn from a binomial population.
2. Independent Individuals – the subjects in the sample are independent of one another and do
not impact the success of one another (you have to think about this)
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3. Success/Failure Condition – n1 p1 and n1 1  p1   10 , shows that n is large enough.
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4. 10% Condition – the size of each sample is less than 10% its population.
SHOW: This is when you calculate a confidence interval. pˆ  z *
pˆ  qˆ 
n
TELL: You draw a conclusion in context: “I am ____ % confident that the proportion of all
___________________ is between ________ and _________.
Example: Suppose you have an SRS of 40 buses from a large city and find that 24 have a safety
violation. Find a 95% confidence interval for the proportion of all buses that have a safety violation.
THINK:
Conditions:
1. Random Condition - The context indicates that it is a random sample and it is
binomial because they either have a safety violation or they do not.
2. Independence- we assume that one buses safety violation status would not impact
another’s therefore each bus should be independent of the other buses.
3. Success/Failure Condition - npˆ  24(1) = 24 and n  qˆ   40(.4)=16 are both at least
10, so n is large enough.
4. 10% Condition – In a large city as indicated by the context the number of buses
would be larger than 40(10) = 400.
SHOW:
pˆ  z
pˆ  qˆ 
.6 .4 
 .6  1.96
 .6  .0775 or about (0.5225, .6775)
n
40
I am 95% confident that the proportion of all of this city’s buses that have a safety
violation is between 0.53 and 0.68.
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TELL:
Your Turn: A statistics professor asked her students whether or not they were registered to vote. In
a sample of 50 of her students (randomly sampled from her 700 students), 35 said they were registered
to vote. Find a 95% confidence interval for the true proportion of the professor’s students who were
registered to vote. What does it mean to be “95% confident”?