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APSTAT UNIT 4A INFERENCE PART 1

APSTAT UNIT 4A INFERENCE PART 1. APSTAT Chapter 18 Sampling Distributions and Sample Means. Lets Just DO IT!!!!. Proportion of correct answers on last AP Stat Exam. Regular Old Distribution:. .55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .85-.89. Lets Just DO IT!!!!.

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APSTAT UNIT 4A INFERENCE PART 1

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  1. APSTAT UNIT 4AINFERENCE PART 1

  2. APSTAT Chapter 18Sampling Distributionsand Sample Means

  3. Lets Just DO IT!!!! Proportion of correct answers on last AP Stat Exam Regular Old Distribution: .55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .85-.89

  4. Lets Just DO IT!!!! Proportion of correct answers on last AP Stat Exam • Now, Everyone take a 5-person random sample • Do randint(1,13,5) to choose your subjects • Add their scores and divide by 5 to get x-bar (sample mean) • Now we will do a distribution of our sample means – a SAMPLING DISTRIBUTION!!!!!

  5. Lets Just DO IT!!!! Proportion of correct answers on last AP Stat Exam Class Sample Means: Sampling Distribution .55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .85-.89

  6. We Just DID IT!!!! • Give me at least 2 things that are different between the regular distribution and the sampling distributions:

  7. Bias • Unbiased Statistic • Mean of Sampling distribution should equal True population mean. • How did ours look earlier? The true mean of the population was about 72.3….

  8. Sample Proportions • Mean of Sample Proportion • In last section: • Proportion is just outcome divided by n, so….

  9. Standard Deviation of Sample Proportion • In last section: • Proportion is just outcome divided by n, so throw down a little Algebra….

  10. Now try it…Sample Proportions • Find mean and standard deviation for: • 60 samples of 10 coin flips, p=.5 • 60 samples of 50 coin flips, p=.5 • What does this say about variability in regards to sample size?

  11. 2 Rules of Thumb - Assumptions/Conditions • Population size large enough • Population should be at least 10 times the sample size • 10% Condition • Normalness • n should be large enough to produce an approximately normal sampling distribution. • np > 10 AND n(1-p) >10

  12. Try them out • A San Jose firm decide to sample 25 residents to determine if they oppose off-shore oil drilling. They predict that P(oppose) = 0.4 • Large enough population? • Normalness?

  13. Example…. • If the true percentage of students who pass the APStat exam is .64, what is the probability that a random sample of 100 students will have at least 70 students pass?

  14. =.64, n=100, 70 or more • Check Conditions - Briefly Explain • 10% • np and n(1-p) > 10 • Draw Picture- (Find SD too) • Find P-Value • Conclusion

  15. Same Problem Data… • Within what range would we expect to find 95% of sample proportions of size 100.

  16. Sample Means • Take a whole bunch of samples and find the means • Why sample means? • Remember our sample of class scores? • Less variable • More normal

  17. Mean and Standard Deviation of X-BAR • If we take all the possible samples from a population, the mean of the sampling distribution will equal the population mean (if the population mean was accurate in the first place, but more on that later)

  18. Mean and Standard Deviation of X-BAR • Standard Deviation of a sampling distribution is:

  19. Let’s try it! • If adult males have height N(68,2) what would be the mean and standard deviation for the distribution if: • n=10 • n=40 • What happened to the Standard deviation when n was quadrupled? • What would happen to the standard deviation if n was multiplied by 9?

  20. CLT – The Central Limit Theorem • If the population we are sampling from is already normal with N(,), the sampling distribution will be normal as well with mean  and standard deviation • But what if the population we are sampling from is not normal?

  21. Age of Pennies • Riebhoff has 50 pennies, he took the current year and subtracted it from the date on the penny to obtain the following data…

  22. Penny Ages

  23. Sample Size n=1

  24. Sample Size n=4(everyone do 3 SRS)

  25. Sample Size n=8(everyone do 3 SRS)

  26. What happened? • The distribution got “normaler” as the sample size increased. Cool? • Central Limit Theorem says that even if a distribution is not normal, the distribution of the sampling distribution will approach normalcy when n is large. • Allows us to use z-scores and such, even when the larger population is not normally distributed.

  27. Assumptions/Conditions • Random Sample - Always describe • Independence - Describe • 10% Condition

  28. Try it • If the APSTAT EXAM 2005 had a mean score of 3.2 with a standard deviation of 1.2… • Old Skool - Find the probability that a single student would have a score of 4 or higher? • New Skool – find the probability that an SRS of 20 students would have a score of 4 or higher?

  29. Old Skool - Find the probability that a single student would have a score of 4 or higher?

  30. New Skool – find the probability that an SRS of 20 students would have a score of 4 or higher? • Check Conditions - Briefly Explain • 10% • Independence and Random Sample • Draw Picture- (Find SD too) • Find P-Value • Conclusion

  31. Standard Error • Sometimes we do not have the population standard deviation. If we have to estimate it, we call it Standard Error and roll an SE.

  32. APSTAT Chapter 19Confidence Intervals for Proportions

  33. Confidence Interval for a Proportion (aka One-Proportion Z-Interval) • At Woodside High, 80 students are surveyed and 32% of them had tried marijuana. • How confident am I that the true proportion of WH students that have tried marijuana is at or near 32%? • CONFIDENCE INTERVAL!!!

  34. The Dealio… • If I do know the population mean • If I sample, I know the sample mean might be quite different than the population mean • BUT…That difference is predictable. • For instance, if N(0.70,0.1) and n=4 • Sample Mean = 0.70, Sample SD=0.1/sqrt4=.05 • We expect 95% of samples (Empirical Rule) to fall between 2 SD of the mean • Therefore 95% of samples will fall between 0.6 and 0.8.

  35. Confidence Intervals • Work in reverse • (From Woodside High Example) I sampled 80 and got sample p = .32 • I want to know the true population proportion. • The true population proportion will lie within 2SD of the Sample Proportion in 95/100 samples of this size. • Let’s Do It!!!!

  36. Do It! • List what you know • p-hat=.32, n=80 • Conditions/Assumptions • 10% for Independence • Woodside HS has over 800 students • np and nq > 10 to use Normal Model • Both .32 x 80 and .68 x 80 > 10 • Find Standard Error • SE(p-hat)=

  37. Do It! • Draw the Picture • Conclusion: • We are 95% confident that the TRUE mean proportion of ____________ falls between ____ and ____

  38. Do It! • We can also write confidence intervals in the form: (estimate) ± (margin of error) Standard error

  39. What Does 95% Confidence Mean? • If we did a whole bunch of confidence intervals at this sample size, we would expect 95 out of 100 intervals to contain the true mean. • Picture of this: TRUE POPULATION PROPORTION

  40. AHOY! • We do not always want 95% confidence. • Example, if a part on an airplane’s landing gear needs to be a certain size to work, wouldn’t you want a little more confidence in the sample being within certain parameters? • Common Intervals are 90, 95 and 99% • Denote as C=.90, C=.95, or C=.99

  41. Area = 90% p But 90 and 99% aren’t Empirically Cool • We need this z-score! It’s critical! • So critical, it is called the critical value and denoted as z*

  42. Mas z* • Now check t distribution critical values chart (back of book or formula sheet) • Look at bottom. It gives you C and right above it is….. • Yeah!

  43. Try it! • A poll asked who would you vote for if an election were held today between Sen. Barack Obama and Sen. John McCain. 115 of the 250 respondents chose Sen. McCain. Construct and interpret a 90% confidence interval for the proportion of voters choosing McCain.

  44. Try It! • Conditions: • Mean, SE, z* • Calculate CI • Conclusion

  45. Last thing • Finding sample size needed for a CI with a given level of confidence and a given margin of error • NBC News is doing a poll on who will be the next Governor of California. The want a 3% margin of error at a 95% confidence interval. What sample size should they use?

  46. Sample size needed Margin of Error

  47. Sample size needed Why 0.5? Gives us largest n value. Safety First! OOPS! YOU SHOULD ALWAYS ROUND UP TO STAY WITHIN CONFIDENCE INTERVAL! SHOULD BE 1068.

  48. APSTAT Chapter 20One ProportionHypothesis Tests

  49. Significance Tests • Example. AP Stat Exam 2005: • National Proportion Who Passed = .58 • Priory Students n = 32, p-hat=.78 • Two Possibilities • Higher WPS proportion just happened by chance (natural variation of a sample) • The likelihood of 78% of 32 students passing is so remote we must conclude that Priory Students are likely better at APStat than national average.

  50. Hypothesis Testing • Reflect our two possibilities from above: • NOTHING IS STRANGE (difference could have been by natural variation of sample) • SOMETHIN’ IS GOIN’ ON (difference is so improbable we must assume there is a difference) • Here is how we write them: • H0: Null Hypothesis (Nothing Strange) • Ha: Alternative Hypothesis (Somethin’ is goin’ on)

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