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Section 8.1

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.

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Section 8.1

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  1. Section 8.1 Confidence Intervals for a Population Mean

  2. 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

  3. Confidence Intervals • Allow you to estimate the value of a population parameter.

  4. 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?

  5. What can we come up with? • We are looking at sample means, so the distribution is approximately….. • Normal by… • CLT • What is the standard deviation of x-bar? Assume that the standard deviation for the population is 100. • 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?

  6. 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, and is found by multiplying Critical Value(z* or t*) times your SD. • The confidence level, C, gives the probability that the interval will capture the true value (μ) in repeated samples (ex. 95% confidence interval)

  7. PANIC • P: Parameter of interest- Define it • A: Assumptions/conditions • N: Name the interval • I: Interval (confidence) • C: Conclude in context

  8. P: Parameters • Parameter: the statistical values of a population (represented by a Greek letter) • Define in first step of confidence interval • µ= the true mean summer luggage weight for Frontier Airline passengers

  9. A: Assumptions • The data comes from an SRS from the population of interest. • The sampling distribution of x bar is approximately normal. (Normality). • By central limit theorem if sample greater than 30 • By graphing in your calculator if you have data • Individual observations are independent; when sampling without replacement, the population size N is at least 10 times the sample size n. (Independence).

  10. N: Name Interval • Interval: Z-interval for Population means

  11. I: Interval • Est. +/- Critical Value*SD • Therefore our Population Mean should be between 452.23 and 469.76

  12. C: Conclude in Context • We are 95% confident that the true mean SAT Math score amongst Seniors in California is between 452.23 and 469.76. • We are __% confident that the true mean [context] lies between (____,____).

  13. 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 68-95-99.7? • 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.

  14. Common Confidence Levels

  15. 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.

  16. Example • Here are measurements (in mm) of a critical dimension on a sample of auto engine crankshafts: 224.12, 224.01, 224.02, 223.98, 223.99, 223.96, 223.96, 224.09, 223.99, 223.98, 223.90, 223.98, 224.1, 224.06, 223.91, 223.99 • 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.

  17. 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?

  18. Oh, behave! • How do Confidence Intervals behave? • Let’s look at the margin of error portion of the formula. • What happens when the sample size increases? • What about the Confidence Level increasing? • What happens when σ gets smaller?

  19. Homework Confidence Interval WS

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