Chapter 11

1. Chapter 11 Day 1

2. Millie Robinah Kenji Sam Empty NEW SEATS!!! 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 NEW SEATS!!! 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 Colin Kerry Tom Kat NEW SEATS!!! Rachel H. Mikki Kim Callie Ayeman Nicky Ben Lucy Empty Jacob Harper Alex Sam Owen Abbie Kate Rachel K DOOR

5. Pre-Assessment • Work on this for 10 minutes • If you finish early, work on the warm up: • 11.15, 11.16 in book.

6. Pass out guided notes • Follow along with the guided notes.

7. 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?

8. How do the z and t Distributions Compare? Similarities Differences 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 • symmetric about zero • single-peaked • bell-shaped

9. 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

10. 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.

11. 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 .

12. 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.

13. 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 .

14. 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.

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

17. 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:

18. 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!

19. 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.

20. 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

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

22. Exit Ticket • What is t* for a 90% confidence interval if degrees of freedom =10? • As the degrees of freedom increase, the t distribution approaches the ___________ distribution. • The weights of 9 men have mean x-bar = 175 pounds and standard deviation s = 15 pounds. The population average is 170 pounds? • What is the standard error of the mean? • What is the one-sample t statistic for the data in problem 3? • What two upper tail probabilities does this t statistic fall between?