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Section 8.1 Confidence Intervals for a Population Mean What has been the point of all this? The point – Inference! The whole point of statistics is so we can “infer” information about our population from our sample data. Statistical inference – methods for drawing conclusions about a population from sample data Confidence Intervals Allow you to estimate the value of a population parameter. Let’s start off easy Suppose you want to estimate the mean SAT Math score for the more than 350,000 high school seniors in California who take the SAT. You look at a simple random sample of 500 California high school seniors who took the SAT. The mean of your sample is x-bar = 461. What can we say about μ for the population of California seniors? What can we come up with? We are looking at sample means, so the distribution is approximately….. What is the standard deviation of x-bar? Assume that the standard deviation for the population is 100. Normal by… CLT 4.5 We know that x-bar should be fairly close to μ. If we want to know with 95% confidence, then we want to know what values μ could be between so that there is 95% area between. Does the 95% sound familiar? Confidence Interval Confidence interval = estimate ± margin of error Estimate is the x-bar in this example (what you get from your sample) Margin of error is how far we are willing to go from the estimate (9 in the last example) The confidence level, C, gives the probability that the interval will capture the true value (μ) in repeated samples (ex. 95% confidence interval) Steps to Construct a Confidence Interval 1) Identify the population of interest and the parameter (μ, p, or σ) you want to draw conclusions about. 2) Verify the conditions for the selected confidence interval. 3) If the conditions are met, use the formula for a CI: CI = estimate ± margin of error 4) Interpret your results in the context of the problem. Step 1: Population of Interest and Parameters In this section, our parameter of interest is a population mean, or μ. We will use x-bar as our estimate of μ. Find x-bar if it is not given to you. Step 2: Conditions There are conditions that MUST be satisfied before you construct a confidence interval. The data must be a SRS from the population of interest. They will tell you this in the question. 2) The sampling distribution of x-bar is approximately normal. If we’re given the actual data, we can plot it to see if the distribution is approximately symmetric (remember normal probability plots). Otherwise, we can assume that the distribution is normal if n is large (CLT). The sampling distribution is approximately normal if the population distribution is approximately normal 1) Step 3: The Formula The general formula for a Confidence Interval is CI = estimate ± margin of error. Specifically, the CI formula for μ is Our unbiased estimator of μ. xz n * Z* will vary based on the Confidence Level. This is the standard deviation of the sampling distribution of xbar. Z* We know that for our interval to have 95% confidence, we should go out 2 standard deviations from x-bar. What about levels of confidence other than 6895-99.7? Draw a picture. Shade the middle region. Find the area TO THE LEFT of z*. Look this value up in the BODY of Table A. Find the Z score that corresponds to that area. OR You can check for common z* upper tail values by looking at the bottom of Table C. Common Confidence Levels Confidence Level Tail Area 90% .05 95% 99% Z* Step 4: Express your results in CONTEXT Fill in the blanks…. We are (insert confidence level) confident that the true (mean or other parameter) of (put in your context) is between (lower bound) and (upper bound). If you forget this, you can find it at the end of your book. Example Here are measurements (in mm) of a critical dimension on a sample of auto engine crankshafts: 224.120, 224.001, 224.017, 223.982, 223.989, 223.961, 223.960, 224.089, 223.987, 223.976, 223.902, 223.980, 224.098, 224.057, 223.913, 223.999 The data come from a production process that is known to have standard deviation σ = 0.060mm. The process mean is supposed to be μ = 224 mm but can drift away from this target during production. Give a 95% confidence interval for the process mean at the time these crankshafts were produced. You try this one A hardware manufacturer produces bolts used to assemble various machines. Assume that the diameter of bolts produced by this manufacturer has an unknown population mean 𝝁 and the standard deviation is 0.1 mm. Suppose the average diameter of a simple random sample of 50 bolts is 5.11 mm. Calculate the margin of error of a 95% confidence interval for 𝝁 Find the 95% Confidence Interval for 𝝁. What is the width of a 95% confidence interval for 𝝁? Oh, behave! How do Confidence Intervals behave? Let’s look at the margin of error portion of the formula. margin of error = z n * What happens when the sample size increases? What about the Confidence Level increasing? What happens when σ gets smaller? Homework Confidence Interval WS