Introduction

In the earlier section (Calculating confidence intervals), we have seen how to estimate population mean from sample mean and calculating confidence intervals around the sample mean. In this section, we will understand calculating confidence intervals for estimating population proportion from sample.

Terminology

  • Population proportion p
  • Proportion q = 1-p
  • Sample proportion prop_3-0
  • Sample size n

Normal approximation for distribution of sample proportion

If both n*p and n*(1-p) are greater than 10, distribution of sample proportion could be assumed as normal. Sample proportion has to be used if the population proportion is not available

Standard error of proportion

Standard deviation associated with the calculation of proportion is called standard error.

prop_ci_2-0

Confidence intervals for proportion

Confidence intervals for estimating population proportion from sample is given by below equation.

prop_ci_1-0

Example

A researcher wants to analyze the proportion of stocks that give positive returns on NYSE in a particular time frame (say one year). She randomly picks up 70 stocks and found that 35% of them gave positive return over past year. If the researcher wants to specify the proportion of stocks that yield positive returns. What is the interval she needs to specify if she wants to be 99% confident?

Solution

Sample size n = 70

Sample proportion = 0.35

n*p  = 70*0.35 and

n*(1-p) = 70*0.65

both are greater than 10. Hence, normal approximation could be used.

Standard error in estimating proportion is given by

prop_ci_2-0

Standard error = 0.057

Confidence interval = Sample proportion +/-  Z * std. error

prop_ci_1-0

= 0.35 +/- 2.57*0.057

Confidence interval

  • Lower limit = 0.203
  • Upper Limit = 0.496

Hence, the limits that researcher should specify for the proportion of stocks that yield positive returns are 20.3% and 49.6%