Confidence Interval (CI) for the Mean When σ Is Known MM4D2 Using student-generated data from random samples of at least 30 members, students will determine the margin of error and confidence interval for a specified level of confidence.

Definition A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate. The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated.

Point Estimate A point estimate is a specific numerical value estimate of a parameter. The best point estimate of the population mean µ is the sample mean

Calculating critical values Z* or Zα/2 Z* or Zα/2 = 1 – { (1- CL decimal form)/2} Look for the closest number to that value in the body of the z-score table Then write down the number to the far left and all the way at the top of that value. Find the critical value of the following: a) 92% = b) 94% = c) 98% =

Critical value solution 92% ≈ 1.75 94% ≈ 1.88 98% ≈ 2.33

Common critical values Z* or Zα/2  

Margin of error ME  

Distribution of

Formula for the Confidence Interval of the Mean for a Specific α When σ is Known    

Example The following data represent a sample of the assets (in millions of dollars) of 30 credit unions in southwestern Pennsylvania. Find the 90% confidence interval of the mean. 12.23 16.56 4.39 2.89 1.24 2.17 13.19 9.16 1.42 73.25 1.91 14.64 11.59 6.69 1.06 8.74 3.17 18.13 7.92 4.78 16.85 40.22 2.42 21.58 5.01 1.47 12.24 2.27 12.77 2.76

Example Solution  

Solution continues  

Instructions for finding the mean and standard deviation for a TI-83/84 http://www.tc3.edu/instruct/sbrown/ti83/dstats.htm