Section 8.2 - Confidence Intervals for Population Proportions
Before constructing a confidence interval for a population proportion, check the conditions:
1. Random: The data come from a well-designed random sample or randomized experiment.
1
 10%: If sampling without replacement, check 𝑛 ≤ 10 𝑁.
2. Large Counts: 𝑛𝑝̂ ≥ 10 and 𝑛(1 − 𝑝̂ ) ≥ 10
**Using 𝑝̂ because we don’t know p. If we knew p, we wouldn’t be trying to estimate it.
If one of the conditions is violated, there is no point in constructing a confidence interval for p.
**Standard Error
When the standard deviation of a statistic is estimated from data, the result is called the standard error of the
statistic.
𝑝̂(1−𝑝̂)
Standard error of p̂ (or standard deviation of p̂ ) = 𝜎𝑝̂ = 𝑆𝐸 = √
𝑛
**Critical Value
If the Large Counts condition is met, we can use a Normal curve. We will use the notation z* to represent the
critical value. The z* represents the endpoints of a C% confidence interval such that C% of its area is between
-z* and z*.
Two ways to find confidence levels:
 Look in standard Normal chart
 Use calculator
Example, p. 497, 80% confidence
Use Table A or technology to find the critical value z* for an 80% confidence interval. Assume that the Large
Counts condition is met.
 Find 1 – C: 1 – 0.80 = 0.2
 Divide by 2: 0.2 / 2 = 0.1
 Look this proportion up in the table, and work backwards to find the value of z* OR in your calculator:
invNorm(0.1) Look at calculator instructions sheet.
z = -1.28 is the closest, so we will have the critical value z* = 1.28.
Common Confidence Levels that you should probably memorize:
Confidence Level
Tail Area
z*
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576
Steps for constructing a Confidence Interval:
 State – know what parameters we’re estimating & at what confidence level
 Plan – choose method & check conditions
 Do – if conditions are met, perform calculations
 Conclude – interpret the interval in the context of the problem
Statistic
Problems
Demand
Consistency
One-Sample z-interval for a Population Proportion:
Draw an SRS of size n from a large population with unknown proportion p of successes. An approximate level
C confidence interval for p is
𝑝̂ ± 𝑧 ∗ √
𝑝̂ (1 − 𝑝̂ )
𝑛
where z* is the upper (1 – C) / 2 standard normal critical value.
Example, p. 500
The Gallup Youth Survey asked a random sample of 439 U.S. teens aged 13 to 17 whether they thought young
people should wait to have sex until marriage. Of the sample, 246 said “Yes.” Construct and interpret a 95%
confidence interval for the proportion of all teens who would say “Yes” if asked this question.
246
Will need 𝑝̂ = 439 ≈ 0.56.
State: We want to estimate the true proportion p of all 13- to 17-year olds in the U.S. who would say that
young people should wait to have sex until they get married with 95% confidence.
Plan: One-sample z –interval for p
Check Conditions: Random: Random sample of U.S. teens
1
 10%: Sampling without replacement, so 439 ≤ 10 (U.S. teens aged 13 – 17).
There are at least 4,390 U.S. teens between 13 and 17 years old.
Large Counts: (439)(0.56) ≈ 246 ≥ 10 and (439)(0.44) ≈ 193 ≥ 10. Satisfied.
246
Do: 𝑝̂ = 439 ≈ 0.56
CL: 95% so z* = 1.96
𝑝̂ ± 𝑧 ∗ √
n = 439
𝑝̂ (1 − 𝑝̂ )
(0.56)(0.44)
= 0.56 ± 1.96√
𝑛
439
= 0.56 ± 0.046
= (0.514, 0.606)
Conclude: We are 95% confident that the interval from 0.514 to 0.606 captures the true proportion of 13to 17-year olds in the U.S. who would say that teens should wait until marriage to have sex.