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.