Unit 4 Topic 3: Confidence Intervals and Sample Size for Proportions
Symbols Used in Proportion Notation
p = population proportion
= sample proportion (pronounced p hat)
For sample proportion:
Where X = number of sample units that possess the characteristics of interest and n = sample size
Proportion Notation
In a recent survey of 150 households, 54 had central air conditioning. Find andwhereis the proportion of households that have central air conditioning.
Confidence Intervals for Proportions
- Confidence Intervals about proportions must meet the criteria that
- Formula for a Specific Confidence Interval for a Proportion:
Example 1
- A sample of 500 nursing applications included 60 from men. Find the 90% confidence interval of the true proportion of men who applied to the nursing program.
Example 2
- A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore.
Sample Size for Proportions
- To find the minimum sample size for a population proportion, we solve the maximum error of the estimate value for n in the formula
- We get
- If necessary, round up to obtain a whole number.
Sample Size for Proportions
Two Scenarios to Consider
- Known
- When we know p hat, we can use it in the formula
- Unknown
- When we don’t know p hat, use
- This value will give a sample size sufficiently large to guarantee an accurate prediction, given the confidence interval and the error of estimate.
- The product of p hat and q hat will be at a maximum, 0.25.
Example 1
- A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary.
Example 2
- The same researcher wishes to estimate the proportion of executives who own a car phone. She wants to be 90% confident and be accurate within 5% of the true proportion. Find the minimum sample size necessary.