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Chapter 11

Chapter 11. Day 2: 2/27, 2/28. Millie Robinah Kenji Sam. Empty. Empty. Juan Collin Jesse Erica. Romel Isabel Gabby. Empty. Liz Anna Reach Healey. Cailtin Drew Alyssa Rehema. DOOR. Ashlie Blake Ben S. Kaushik. Jeremy Claire Allison S. Ben W.

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Chapter 11

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  1. Chapter 11 Day 2: 2/27, 2/28

  2. Millie Robinah Kenji Sam Empty Empty Juan Collin Jesse Erica Romel Isabel Gabby Empty Liz Anna Reach Healey Cailtin Drew Alyssa Rehema DOOR

  3. Ashlie Blake Ben S. Kaushik Jeremy Claire Allison S. Ben W. Daniel Lindsey Lauren Kadey Carrie Cassidy Zoe Allison M. ShelbyMaggie Ian Aikera Empty Simon Robby Juan Jenny Tyreic Anna Carling Jessica DOOR

  4. Thomas Elliot Nick Alandra Deaja Wynton Michael Alexis Lucy Kerry Tom Kat Rachel H. Mikki Kim Callie Ayeman Nicky Ben Colin Empty Jacob Harper Alex Sam Owen Abbie Kate Rachel K DOOR

  5. Pass out guided notes Follow along with the guided notes.

  6. Essential Question: How do we construct a C level confidence interval or perform a test of significance for the population mean if we aren’t given σ?

  7. Differences How do the z and t Distributions Compare? Similarities symmetric about zero single-peaked bell-shaped t distribution has larger spread (more probability in tails, less in center) as the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve ever more closely

  8. Chapter 11 is all about inference for the mean of a population. • Next chapter, we will look at inference for the proportion of a population. • This chapter describes confidence intervals and significance tests for the mean of a single population and for comparing two populations

  9. Conditions for Inference about a Mean • We get our data from an SRS of size n from our population • Observations from the population have a normal distribution with a mean and a standard deviation , both of which are unknown • In practice, it is enough that the distribution is symmetric and single-peaked unless the sample is very small.

  10. Standard Error Just like we have seen before, the sampling distribution of the sample mean is approximately normal with mean and standard deviation . BUT, we do not know what is. So what do we do? We estimate using the sample standard deviation Therefore, we now describe the sample distribution of as approximately normal with mean and standard deviation . The standard error is .

  11. A new distribution • Like we saw in Chapter 10, when we know , we base our inference on the one-sample z statistic . • The z statistic has distribution • However, we do not know . So, we substitute the standard error of for its standard deviation. • We now have, . • This is not normally distributed. Instead, it has a new distribution called…. • The distribution.

  12. The one-sample statistic and the distribution. Draw an SRS of size n from a population that has the normal distribution with unknown mean and standard deviation. The one-samplestatistichas the distribution with n-1 degrees of freedom. For short, we write the distribution with degrees as .

  13. distribution • The density curves of the distribution are similar in shape to the standard normal curve. • Both symmetric about zero, single-peaked, and bell-shaped. • The distribution has a greater spread. • There is more probability in the tails and less in the center • This follows because the estimate introduces more variation into the statistic than the known parameter • As the degrees of freedom increases, the density curve does approach the standard normal density curve • This follows because estimates better as n increases. As n increases, so do the degrees of freedom. • Table C gives the critical values for the distribution.

  14. degrees of freedom P-values values of t*

  15. The One-Sample t Procedures A level C confidence interval for μ is: To test the hypothesis H0: μ = μ0, compute the one-sample t statistic:

  16. Inference procedures for distribution. • Same process as Chapter 10, just replacing with the standard error • Confidence Intervals: • Significance Tests • vs: • is • is • is • These P values are exact if the population distribution is normal and approximate for large n in other cases. DO NOT FORGET YOUR INFERENCE TOOLBOX!

  17. Robustness • The confidence interval and test are exactly correct when the distribution of the population is exactly normal. • This never happens with real data. No real data is exactly normal • The usefulness of the procedures therefore depend on how strongly they are affected by the lack of normality. • A confidence level or significance test is called robust if the confidence level or P-value does not change very much when the assumptions of the procedure are violated. • The procedures are strongly influenced by outliers • However, the procedures are robust against nonnormality of the population when there are no outliers. • Always make a plot to check for skewness and outliers before using the procedures for small samples.

  18. Using the procedures • Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more important than the assumption that the population distribution is normal • Sample Size less than 15: • Use procedures if that data are close to normal. If that data are clearly nonnormal or if outliers are present, do not use • Sample size at least 15: • The procedures can be used except in the presence of outliers or strong skewness • Large samples: • The procedures can be used even for clearly skewed distributions when the sample is large, roughly

  19. “Introduction to the Student’s t-distribution” Work on this with your group for 15 minutes.

  20. Introduction to Matched Pairs Work on #11.31 & 32

  21. Matched Pairs procedures To compare the responses to the treatment in a matched pairs design, apply the one-sample procedures to the observed differences. The parameter in a matched pairs procedure is the mean difference in the responses to the two treatments within matched pairs of subjects in the entire population.

  22. Matched Pairs Comparative studies are more convincing than single-sample investigations. One common design to compare two treatments makes use of one sample procedures In matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair. We take the difference of each match pair to produce a single sample.

  23. The power of the test As we saw in Chapter 10, the power of a statistical test measures its ability to detect deviations from the null hypothesis. We carry out the test in hopes that the null hypothesis is false, so a high power is preferable. The power of the one-sample test against a specific alternative value of the population mean is the probability that the test will reject the null hypothesis when the mean has this alternative value To calculate this probability, we need a fixed level of significance, usually

  24. AP Stat Free Response Question • In your group, work on the example free response question from a past AP test. • Work on it for 10 minutes with your group. • Remember the Inference Toolbox • This is your exit ticket today!!!!

  25. Solutions

  26. Homework • What did we look at today? • Matched-pairs testing • Do # 20, 38, 42, 50, 53 • Read and take notes on section 11.2

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