Reminders. Quiz today - please bring a calculator I ll post the next HW by Saturday (last HW!)

Transcription

1 Reminders Quiz today - please bring a calculator I ll post the next HW by Saturday (last HW!) 1

2 Warm Up Chat with your neighbor. What is the Central Limit Theorem? Why do we care about it? What s the (long) interpretation of a confidence interval? 2

3 Warm Up - Interpretation Why do they have different centers? Different intervals? % CIs from Samples of size 17 assuming p=0.25 Sample # p 3

4 Chapter 21 Homework and Review Problems Aaron Zimmerman STAT Summer 2014 Department of Statistics University of Washington - Seattle 4

5 Road Map for Confidence Interval Problems Am I calculating a confidence interval for mean or proportion? Proportion Mean Find or calculate the sample proportion, ˆp. Find the sample size n Calculate the standard error: ˆp(1 ˆp) Find the critical value z for your confidence level C Calculate the CI: ˆp(1 ˆp) ˆp ± z n n Find or calculate the sample mean, x and the sample standard deviation s. Find n. Calculate the standard error: s n Find the critical value z for your confidence level C Calculate the CI: x ± z s n 5

6 Practice A box contains a large number of red and blue marbles, but the proportions are unknown; 100 marbles are drawn at random, and 53 turn out to be red. Say whether each of the following statements is true or false, and explain briefly. The percentage of red marbles in the box can be estimated as 53%; the SE is 5%. The 5% measures the confidence in the 53%. The 53% is likely to be off the percentage of red marbles in the box, by 5% or so. A 95%-confidence interval for the percentage of red marbles in the box is 43% to 63%. A 90%-confidence interval for the percentage of red marbles in the sample is 48% to 58%. 6

7 Practice True or false: with a well-designed sample survey, the sample percentage is very likely to equal the population percentage. Explain. 7

8 Practice Five hundred draws are made at random with replacement from a box with 10,000 tickets. The average of the box is unknown. However, the average of the draws was 71.3, and their SO was about 2.3. True or false, and explain: The 71.3 estimates the average of the box, but is likely to be off by 0.1 or so. A 68%-confidence interval for the average of the box is 71.3 ± 0.1. About 68% of the tickets in the box are in the range 71.3 ±

9 PISA Mathematics Exam Every three years, the Programme for International Student Assessment (PISA) gives a math, science, and reading exam to a SRS of 15-year-olds within 38 developed countries. We will analyze country-level data from the 2009 mathematics exam I have also broken down the exam scores for the United States by race/ethnicity 9

10 PISA Mathematics Exam 10

11 PISA Mathematics Exam 11

12 PISA Mathematics Exam Look at the column labeled Mean PISA Math Score on the first chart. What is the mean of the sampling distribution for each of these statistics? The sample mean X The population parameter µ The standard error σ n The margin of error z σ n 12

13 PISA Mathematics Exam Look at the column labeled first chart with Mean PISA Math Score. What is the mean of the sampling distribution for each of these statistics? The sample mean X The population parameter µ The standard error σ n The margin of error z σ n 13

14 PISA Mathematics Exam We will calculate 95% confidence intervals for the mean score of students in Portugal and Ireland who passed the exam. Before we do, which confidence interval will be narrower? PORTUGAL sample mean score: 487 sample standard deviation: 56 sample size: 724 IRELAND sample mean score: 487 sample standard deviation: 56 sample size:

15 PISA Mathematics Exam What s the confidence interval for the mean score in Portugal? What about in Italy? PORTUGAL sample mean score: 487 sample standard deviation: 54 sample size: 724 IRELAND sample mean score: 487 sample standard deviation: 56 sample size:

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17 PISA Mathematics Exam Look at the third chart and the bars labeled Below Level 2. These are percentage of exam takers who scored below this level. What is the mean of the sampling distribution for each of these statistics? The sample mean ˆp The population parameter p The standard error The margin of error z p(1 p) n p(1 p) n 17

18 PISA Mathematics Exam We will calculate 90% and 99% confidence intervals for the percent who score below level 2 in Korea. Which one will be narrower? KOREA sample proportion below level 2: 9% sample size: 476 Conf Lvl Crit. Val Conf Lvl Crit. Val 50% % % % % % % % 3.29 Table: Replicated Table 21.1 from book 18

19 PISA Mathematics Exam What s a 90% and 99% confidence intervals for the percent who score below level 2 in Korea. KOREA sample proportion below level 2: 9% sample size: 476 Conf Lvl Crit. Val Conf Lvl Crit. Val 50% % % % % % % % 3.29 Table: Replicated Table 21.1 from book 19

20 PISA Mathematics Exam What s a 95% CI for the PISA Math passing rate in Ireland? What about in Italy? ITALY proportion passed: 83% sample size: 724 IRELAND proportion passed: 83% sample size:

21 What does this mean? If the distribution of the variable in the population is Normal with mean µ and standard deviation σ, then the sampling distribution of x is also Normal with mean µ and standard deviation σ n Distribution of Math ACT Scores mean=18 sd=6 Sampling Distribution of Sample Mean, n=20 mean=18 sd=

22 What does this mean? If the distribution of the variable in the population is Normal with mean µ and standard deviation σ, then the sampling distribution of x is also Normal with mean µ and standard deviation σ n Distribution of Math ACT Scores mean=18 sd=6 Sampling Distribution of Sample Mean, n=50 mean=18 sd= So, when we don t have many observations, we don t know a lot about where µ is, but with more observations, the sampling distribution becomes narrower. 22

23 What does this mean? If the distribution of the variable in the population has any distribution mean µ and standard deviation σ, then with large enough n the sampling distribution of x is still Normal (!!!) with mean µ and standard deviation σ n Skewed Distribution Skewed Distribution

24 What does this mean? If the distribution of the variable in the population has any distribution mean µ and standard deviation σ, then with large enough n the sampling distribution of x is still Normal (!!!) with mean µ and standard deviation σ n Skewed Distribution Sampling Distribution of Sample Mean, n=

25 What does this mean? If the distribution of the variable in the population has any distribution mean µ and standard deviation σ, then with large enough n the sampling distribution of x is still Normal (!!!) with mean µ and standard deviation σ n Skewed Distribution Sampling Distribution of Sample Mean, n=

26 What does this mean? If the distribution of the variable in the population has any distribution mean µ and standard deviation σ, then with large enough n the sampling distribution of x is still Normal (!!!) with mean µ and standard deviation σ n Skewed Distribution Sampling Distribution of Sample Mean, n=

27 What does this mean? If the distribution of the variable in the population has any distribution mean µ and standard deviation σ, then with large enough n the sampling distribution of x is still Normal (!!!) with mean µ and standard deviation σ n Skewed Distribution Sampling Distribution of Sample Mean, n= As n increases, the sampling distribution gets narrower and more Normal. 27

28 Central Limit Theorem The information on the previous three slides is all summarized by (arguably) the most important theorem in statistics: the central limit theorem Central Limit Theorem: In words, the central limit theorem says that if I take enough random draws from any population, the sampling distribution of the sample mean will be Normal, centered at the true mean of the population, and with a standard deviation that s the standard deviation in the population divided by the square root of the sample size. 28

29 Central Limit Theorem Activity (already done) Proportions from samples of size 10 Frequency Average Estimated Statistic True Parameter Value estimated proportion Proportions from samples of size 25 Frequency estimated proportion 29

30 Central Limit Theorem Activity (already done) We can think of proportions as averages of 1s and 0s add up 1s and 0s and you just get the number of 1s and then we divide by n, which is also how many 1s and 0s were added together If you squint, the second plot (n=25) looks much more Normal than the first plot (n=10) 30

31 Confidence Intervals for Means Just like with proportions, we don t know the true mean or standard deviation of the population (they are the parameters) So, we estimate the standard error of the sample mean x as s σ n (instead of n ) Following the exact same argument that we used for proportions, we can use the sample mean and standard error to calculate a level C confidence interval for the population mean 31

32 Confidence Intervals for Means Choose a SRS of size n from a large population of individuals having mean µ The means of sample observations is x When n is reasonably large, an approximate level C confidence interval for µ is: x ± z s n Remember, z* is the critical value for confidence level C from Table

33 Example Suppose we want to estimate the mean height of Douglas Fir trees in Mt. Rainier National Park. In a SRS of 25 trees, the mean height is 75 ft. with a standard deviation of 15 ft. What is a 90% confidence interval for the population mean? Important Info x = 75 s = 15 n = 25 z = % CI x ± z s n 75 ± ± 4.92 (70.08, 79.92) 33

34 Your Turn A linguist wants to estimate the mean number of languages spoken fluently by residents of Switzerland. He calculates a sample mean of 2.75 and a sample standard deviation of 1.8 from a SRS of 81 residents of Switzerland. What is a 95% confidence interval for the population mean? What s the interpretation of this interval? 34

35 Your Turn A linguist wants to estimate the mean number of languages spoken fluently by residents of Switzerland. He calculates a sample mean of 2.75 and a sample standard deviation of 1.8 from a SRS of 81 residents of Switzerland. What is a 95% confidence interval for the population mean? What s the interpretation of this interval? Important Info x = 2.75 s = 1.8 n = 81 z = % CI x ± z s n 2.75 ± ± 0.39 (2.36, 3.14) 35

36 Homework All of the HW for the week is on the website in a pdf. The HW after today is: Read Chapter 21, pp To calculate confidence intervals, we will use the equation on p. 466, NOT the equation on p. 460! Do problems 221.3, 21.28b (calculate the sample mean yourself, but use s = 14.27), 21.30, 21.31,