South Dakota School of Mines & Technology Estimation Industrial Engineering

Estimation Estimates for Proportions Industrial Engineering

Estimating a Proportion Suppose we sample 100 circuit boards and find that 8 are defective. We would like to make an inference about the true percentage defective given a sample defective of p = 0.08.

Estimating a Proportion Suppose we sample 100 circuit boards and find that 8 are defective. We would like to make an inference about the true percentage defective given a sample defective of p = 0.08. Recall that for a large sample (n>30) the binomial may be approximated by the normal distribution. We also know that the mean of the binomial is np and the variance is npq. ) , ( npq np N x » , x = # defects in n items

Estimating a proportion ) , ( npq np N x » n x p = ˆ Now if then, or ) , ( ˆ npq np N p n » ) 1 , ( ˆ N npq np p n » -

Estimating a Proportion ) 1 , ( ˆ N npq np p n » - Divide through by n and replace pq by gives p ˆ q ˆ ) 1 , ( / ˆ N n q p » -

Estimating a Proportion ) 1 , ( / ˆ N n q p » - ) / ˆ ( 1 2 a z n q p P £ - =

Estimating a Proportion ) / ˆ ( 1 2 a z n q p P £ - = Miracle 21c occurs n q p z / ˆ 2 a ±

Example Returning to our circuit board example, suppose a sample of 100 boards yields 8% defective. Compute a 90% confidence interval for the true but unknown proportion defective. n q p z / ˆ 2 a ± 100 / ) 92 (. 08 . 645 1 ±

Example Returning to our circuit board example, suppose a sample of 100 boards yields 8% defective. Compute a 90% confidence interval for the true but unknown proportion defective. 0.035 < p < 0.125