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2.6 Confidence Intervals and Margins of Error What you often see in reports about studies… These results are accurate to within +/- 3.7%, 19 times out of 20. Margin of error Confidence level 95% probability that x is somewhere in the range (x – 3.7, x+3.7) Confidence interval Confidence Intervals • x is sample mean • We don’t usually know population mean, (mu) • We can find confidence intervals – Ranges of values likely to be in – E.g., a 95% confidence level has 0.95 probability of containing Notation • = allowed error, or probability of error • (1 – ) = confidence level • z = z-score for that confidence interval 1 2 – E.g. z0.975 is the z-score for a 95% confidence interval Note: the proper notation is actually z 2 Confidence Intervals • A (1 – ) or (1 – ) x 100% confidence interval for , given population standard deviation , sample size n, and sample mean x , represents the range of values x z 1 2 n x z 1 2 x z n What’s this? Population mean/Sample Means • So far, mean of sample = mean of population • Means from different samples of the same population are different • Sample means have normal distribution 2 X N , n Common confidence levels and their associated z-scores Confidence Level Tail size, 2 z-score, z 90% 0.05 1.645 95% 0.025 1.960 99% 0.005 2.576 1 2 Example 1: Drying Times • A paint manufacturer knows from experience that drying times for latex paints have a standard deviation of 10.5 min. The manufacturer wants to use the slogan “Dries in T min.” on its advertising. Twenty test areas of equal size are painted and the mean drying time is found to be 75.4 min. • A) Find a 95% confidence interval for the actual mean drying time of the paint. • B) What would be a reasonable value for T? Example 1 • A) For a 95% confidence level, the acceptable probability error is = 5% = 0.05 z 1 2 z(0.975) 1.960 x z 1 2 n x z 1 2 n 10.5 10.5 75.4 (1.960) 75.4 (1.960) 20 20 70.8 80.0 The manufacturer can be 95% confident that the actual mean drying time is between 70.8 min and 80.0 min. • B) It would be reasonable to advertise “Dries in 80 min.” Margin of Error and Sample Sizes • Consider the confidence interval width, w – E.g. 70.8 < < 80.0, – w = 9.2 • Margin of Error = half the confidence interval width –the maximum difference between the observed sample mean x and the true value of the population mean E z 1 2 w = 2E n Sample Size • We can use this to calculate the minimum sample size necessary for a given confidence level – Often used in opinion polls and other surveys • If sample size too large, waste of resources/time/money • If sample size too small, inaccurate results x z 1 2 Solving for sample size, n: n w 2 z 1 2 n 2z 1 2 n w 2 Notes • Need to know in advance • Estimate it by doing a pre-survey/study • Margin of error decreases as sample size increases, but only to a point Example 2: ISPs • We would like to start an Internet Service Provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 90% sure that the sample mean is within 1 minute of the population mean . Assume that a previous survey of household usage has shown = 6.95 minutes. Example 2 • For a 90% confidence level, the acceptable probability error is = 10% = 0.10 2z z z(0.995) 2 1 2 1.645 E = 1 min w = 2E = 2 min n 1 2 w 2(1.645)(6.95) 2 2 130.7 You would need a sample of about 131 households.