1. AP Statistics Friday, 29 January 2016 OBJECTIVE TSW determine confidence intervals. Yesterday’s tests are not graded. TEST: Continuous Distributions tests are graded.

2. Test Addendum: Continuous Distributions Up to 5 points will be awarded on the Continuous Distributions test. Directions must be followed. –5 decimals for probabilities –3 decimals for other values –Complete sentence answers when indicated. –All work – including calculator key strokes – shown. This is due on Tuesday, 02 February 2016.

3. AP Exam Registration Registration must be done on line. www.TotalRegistration.net/AP/443381 Regular registration: 01/05/2016 – 03/04/2016. Late registration: 03/05/2016 – 03/20/2016 –Additional $10 per test fee added on. Cost per test: –$96/test Can apply for a Cy-Hope scholarship (reduction of $25/test, up to 3 tests). Pick up applications at Counselors’ Corner or at Mr. Hernandez’s office or Ms. Lewis’ office. –$11/test (free/reduced lunch program) Not eligible for Cy-Hope scholarship.

4. Confidence Intervals

5. Rate your confidence 0 - 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.

6. What happens to your confidence as the interval gets smaller? As the interval gets smaller, your confidence goes down. The larger your confidence, the wider the interval.

7. Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach variationBut not always very precise due to variation in the sampling distribution

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

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

10. Confidence level Is the success rate of the method used to construct the interval “Using this method, ____% of the time the intervals constructed will contain the true population parameter.”

11. What does it mean to be 95% confident? 95% chance that  is contained in the confidence interval. ( ? ? ? ) The probability that the interval contains  is 95%. ( ? ? ? ) The method used to construct the interval will produce intervals that contain  95% of the time. ( ? ? ? ) Which is correct?

12. AP Statistics Monday, 01 February 2016 OBJECTIVE TSW determine confidence intervals. ASSIGNMENT DUE DATES –WS Confidence Intervals #1  tomorrow, 02/02/16 –WS Confidence Intervals #2  Wednesday, 02/03/16 Sampling Distributions tests are not graded. AP EXAM REGISTRATION (through 03/04/16) www.TotalRegistration.net/AP/443381

13. Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz*.051.645.0251.96.0052.576 Critical value (z*).05 z*=1.645.025 z*=1.96.005 z*=2.576 90% 95% 99%

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

15. Steps for doing a confidence interval: 1)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)  is known 2)Calculate the interval 3)Write a statement about the interval in the context of the problem (complete sentence).

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

17. Assumptions: 1)Have an SRS of blood measurements 2)Potassium level is normally distributed (given) 3)  known We are 90% confident that the true mean potassium level is between 3.010 and 3.390. 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  = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

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

19. 99% confidence interval? Assumptions: 1)Have an SRS of blood measurements 2)Potassium level is normally distributed (given) 3)  known We are 99% confident that the true mean potassium level is between 2.903 and 3.497.

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

21. How can you make the margin of error smaller? z* smaller (lower confidence level)  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!

22. Example 2: A random sample of 50 JVHS students was taken and their mean SAT score was 1250. (Assume  = 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

23. 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...

24. 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. Suppose that we have this random sample of SAT scores for JVHS students: 950 1130 1260 1090 1310 1420 1190 What is a 95% confidence interval for the true mean SAT score? (Assume  = 105) Example 2.5:

25. AP Statistics Tuesday, 02 February 2016 OBJECTIVE TSW (1) finish viewing the presentation on confidence intervals, (2) turn in WS #1, and (3) work on WS #2. ASSIGNMENTS DUE –WS Confidence Intervals #1  wire basket –WS Test Addendum: Continuous Distributions  black tray ASSIGNMENT DUE TOMORROW –WS Confidence Intervals #2 QUIZ: Confidence Intervals is tomorrow.

26. 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!

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

28. 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.

29. 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

30. 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

31. How to find t* Use Table B for t distributions (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 Can also use invT on the calculator! Need upper t* value with 5% is above – so 95% is below invT(p,df)

32. Assumptions for t-inference Have an SRS from population  unknown Normal distribution –Given –Large sample size –Check graph of data

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

34. Assumptions: Have an SRS of healthy, white males Systolic blood pressure is normally distributed (given).  is unknown We are 95% confident that the true mean systolic blood pressure is between 111.220 and 118.580. For Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.

35. 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.

36. 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)

37. 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.

38. Example 6: Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: 160200220230120180140 13017019080120100170 Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. (Just find the interval.) (126.1618, 189.5524)

39. 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. NOT EQUAL Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than!

40. 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

41. Some Cautions (continued): Outliers can have a large effect on confidence interval Must know  to do a z-interval – which is unrealistic in practice