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Confidence Intervals

Learn about confidence intervals and how they estimate unknown population mean. Explore the relationship between confidence level, margin of error, and precision of estimates. Understand the importance of assumptions and critical values in constructing confidence intervals.

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Confidence Intervals

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  1. Confidence Intervals Chapter 8

  2. Rate your confidence0 - 100 • Name my age within 10 years? • within 5 years? • within 1 year? • Shooting a basketball at a wading pool, will make basket? • Shooting the ball at a large trash can, will make basket? • Shooting the ball at a carnival, will make basket?

  3. What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval. Simulation

  4. Point Estimate • Use a single statistic based on sample data to estimate a population parameter • Simplest approach • But not always very precise due to variation in the sampling distribution

  5. Confidence intervals • Are used to estimate the unknown population mean • Formula: estimate + margin of error

  6. Shows how accurate we believe our estimate is The smaller the margin of error, the more precise our estimate of the true parameter Formula: Margin of error

  7. Confidence level • Is the success rate of the methodused to construct the interval • Using this method, ____% of the time the intervals constructed will contain the true population parameter

  8. .05 .025 .005 Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level tail area z* .05 1.645 .025 1.96 .005 2.576 z*=1.645 z*=1.96 z*=2.576 90% 95% 99%

  9. What does it mean to be 95% confident? • The probability that the interval contains m is 95% • The method used to construct the interval will produce intervals that contain m 95% of the time.

  10. Confidence interval for a population mean: Standard deviation of the statistic Critical value estimate Margin of error

  11. Steps for doing a confidence interval: PANIC!!!!! / State, Plan, Do, Conclude P-arameter: are we estimating something about means, proportions, slopes? A-ssumptions • SRS from population • Sampling distribution is normal (or approximately normal) • Given (normal) • Large sample size (approximately normal) • Graph data (approximately normal) N-ame Interval I-nterval: Show work C-onclusion: Write a statement about the interval in the context of the problem.

  12. Statement: (memorize!!) We are ________% confident that the interval ______ to ______ captures the true parametercontext.

  13. A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with s = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level? Parameter: true mean level of potassium in the blood Assumptions: • Have an SRS of blood measurements • Potassium level is normally distributed (given) • s known Name the interval: one-sample z-interval Interval: Conclusion: We are 90% confident that the interval 3.01 to 3.39 captures the true mean potassium level.

  14. 95% confidence interval? Parameter, Assumptions, Interval is the same as previous Wider Interval due to larger confidence Interval: Conclusion: We are 95% confident that the interval from 2.974 to 3.426 captures the true mean potassium level.

  15. 99% confidence interval? We are 99% confident that the interval from 2.902 to 3.497 captures the true mean potassium level.

  16. What happens to the interval as the confidence level increases? the interval gets wider as the confidence level increases

  17. Homework • Pg.490 21-24; pg. 504 27, 29, 31, 33, 34

  18. .05 .025 .005 Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level tail area z* .05 1.645 .025 1.96 .005 2.576 z*=1.645 z*=1.96 z*=2.576 90% 95% 99%

  19. How can you make the margin of error smaller? • z* smaller (lower confidence level) • s smaller (less variation in the population) • n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!

  20. A random sample of 50 BGHS students was taken and their mean SAT score was 1250. (Assume s = 105) What is a 95% confidence interval for the mean SAT scores of BGHS students? We are 95% confident that the true mean SAT score for BGHS students is between 1220.9 and 1279.1

  21. If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Find a sample size: Always round up to the nearest person!

  22. The heights of BGHS male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval? n = 43 Homework pg. 489 1, 2, 9-19 odd, 25, 26

  23. 8.3 t- distribution • Developed by William Gosset • Continuous distribution • Unimodal, symmetrical, bell-shaped density curve • Above the horizontal axis • Area under the curve equals 1 • Based on degrees of freedom

  24. Graph examples of t- curves vs normal curve

  25. How does t compare to normal? • Shorter & more spread out • More area under the tails • As n increases, t-distributions become more like a standard normal distribution

  26. How to find t* Can also use invT on the calculator! Need upper t* value with 5% is above – so 95% is below invT(p,df) • Use Table B for t distributions • Look up confidence level at bottom & df on the sides • df = n – 1 Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2.132 t* =2.145

  27. Formula: Standard deviation of statistic Critical value estimate Margin of error

  28. Assumptions for t-inference • Have an SRS from population • s unknown • Normal distribution • Given • Large sample size • Check graph of data

  29. Robust • An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated. • t-procedures can be used with some skewness, as long as there are no outliers. • Larger n can have more skewness.

  30. Outliers are always a concern, but they are even more of a concern for confidence intervals using the t-distribution • Sample mean is not resistant; hence the sample mean is larger or smaller (drawn toward the outlier)(small numbers of n in t-distribution!) • Sample standard deviation is not resistant; hence the sample standard deviation is larger • Confidence intervals are much wider with an outlier included • Options: • Make sure data is not a typo (data entry error) • Increase sample size beyond 30 observations

  31. A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults. We are 95% confident that the true mean pulse rates of adults is between 70.883 and 74.497 beat per minute.

  32. Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain. The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

  33. Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: 160 200 220 230 120 180 140 130 170 190 80 120 100 170 Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. We are 98% confident that the true mean calorie content per serving is between 126.16 and 189.56 calories.

  34. Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than! A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate? Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.

  35. Some Cautions: • The data MUST be a SRS from the population • The formula is not correct for more complex sampling designs, i.e., stratified, etc. • No way to correct for bias in data

  36. Cautions continued: • Outliers can have a large effect on confidence interval • Must know s to do a z-interval – which is unrealistic in practice

  37. Homework: • Pg.528 69, 71, 73-78

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