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AP Statistics Monday , 03 February 2014

AP Statistics Monday , 03 February 2014. OBJECTIVE TSW determine confidence intervals. READ Sec. 9.1: pp. 475-480 Sec. 9.3: pp. 495-496 Triple Quizzes are not graded. Confidence Intervals. Chapter 9. Rate your confidence 0 - 100. Name my age within 10 years. within 5 years.

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AP Statistics Monday , 03 February 2014

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  1. APStatisticsMonday, 03 February 2014 • OBJECTIVETSW determine confidence intervals. • READ • Sec. 9.1: pp. 475-480 • Sec. 9.3: pp. 495-496 • Triple Quizzes are not graded.

  2. Confidence Intervals Chapter 9

  3. Rate your confidence0 - 100 • Name my age within 10 years. • within 5 years. • within 1 year. • Shoot a basketball at a wading pool and make a basket. • Shoot the ball at a large trash can and make a basket. • Shoot the ball at a carnival game of chance and make a basket.

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

  5. 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

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

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

  8. 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.”

  9. What does it mean to be 95% confident? • 95% chance that m is contained in the confidence interval. ( ? ? ? ) • 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. ( ? ? ? ) • Which is correct?

  10. .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%

  11. Confidence interval for a population mean (Formula): Standard deviation of the statistic Critical value estimate Margin of error

  12. Steps for doing a confidence interval: • State the assumptions – • SRS taken from population • Sampling distribution is normal (or approximately normal) • Given (normal) • Large sample size (approximately normal) • Graph data (approximately normal) • s is known • Calculate the interval • Write a statement about the interval in the context of the problem (complete sentence).

  13. Statement: (memorize!!) We are ________% confident that the true mean of context lies within the interval ______ and ______.

  14. APStatisticsTuesday, 04 February 2014 • OBJECTIVETSW (1) finish viewing the presentation on confidence intervals, and (2) work on WS #1: Confidence Intervals. • Triple Quizzes are not graded. • Everyone needs a calculator. • SENIOR PANORAMIC PICTURE SALE • (Should be here) Thursday, 02/06/14, at lunch ($25 each). • AP EXAM REGISTRATION (through 03/07/14) www.TotalRegistration.net/AP/443381

  15. Example 1: 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? • Assumptions: • 1) Have an SRS of blood measurements • 2) Potassium level is normally distributed (given) • 3) s known • We are 90% confident that the true mean potassium level is between 3.010 and 3.390.

  16. Assumptions: • 1) Have an SRS of blood measurements • 2) Potassium level is normally distributed (given) • 3) s known • We are 95% confident that the true mean potassium level is between 2.974 and 3.426. 95% confidence interval?

  17. 99% confidence interval? • Assumptions: • 1) Have an SRS of blood measurements • 2) Potassium level is normally distributed (given) • 3) s known • We are 99% confident that the true mean potassium level is between 2.903 and 3.497.

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

  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. Example 2: A random sample of 50 JVHS 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 JVHS students? We are 95% confident that the true mean SAT score for JVHS students is between 1220.9 and 1279.1

  21. How do you find a critical value (z*) for a given confidence level? • Use invNorm on the calculator. • Example: For a 90% confidence level, invNorm(0.95) = 1.644853626 . . . • For an 84% confidence level, invNorm(0.92) = 1.405071561 . . .

  22. Write this on a separate sheet of paper: • Suppose that we have this random sample of SAT scores: • 1130 1260 1090 1310 1420 1190 • What is a 95% confidence interval for the true mean SAT score? (Assume s = 105) • Assumptions: • SRS (given) • The distribution is approximately normal (“boxplot is symmetrical”or“quantile plot is linear”). • σ is given. • We are 95% confident that the true mean SAT score for JVHS students is between 1115.072 and 1270.642.

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

  24. Example 3: The heights of JVHS male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + 0.75 inches with a 95% confidence interval? n = 43

  25. Assignment • WSConfidence Intervals #1 • Change: • #3: Add “Crop researchers plant a random sample of 34 plots …” • This is due at the beginning of class on Thursday, 06 February 2014.

  26. Example 4: In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, giving them either calcium or a placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not? No – the population standard deviation (σ) is not known.

  27. Student’s 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

  28. 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

  29. 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 tdistributions (green chart) • 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

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

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

  32. For Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. • Assumptions: • Have an SRS of healthy, white males • Systolic blood pressure is normally distributed (given). • s is unknown • We are 95% confident that the true mean systolic blood pressure is between 112.3616 and 117.4384.

  33. 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.

  34. Example 5: 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. (Just find the interval.) (70.8834, 74.4965)

  35. 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.

  36. Example 6: 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. (Just find the interval.) (126.1618, 189.5524)

  37. 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 suggests that the average calories per serving does not equal 120 calories.

  38. 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

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

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