Statistics and Quantitative Analysis U4320

1. Statistics and Quantitative Analysis U4320 Segment 5: Sampling and inference Prof. Sharyn O’Halloran

2. Sampling • A. Basics • 1. Ways to Describe Data • Histograms • Frequency Tables, etc. • 2. Ways to Characterize Data •  Central Tendency • Mode • Median • Mean • Dispersion • Variance • Standard Deviation

3. Sampling(cont.) • 3. Probability of Events • If Discrete • Rely on Relative Frequency • If Continuous • Rely on the distribution of events • Example: Standard Normal Distribution • 4. Samples • We can take a sample of the population and make inferences about the population. • 5. Central Question • How well does the sample represent the underlying population?

4. Sampling (cont.) • B. Random Sampling • 1. Problems with Sample Bias • The way we collect our data may bias our results. That is, the average response in our sample may not represent the average response in the whole population. • Examples: • Literary Digest Phone Book Poll • Primaries • Relation between economic growth and education looking only at OECD countries • 2. Solution • Random Sampling

5. Sampling (cont.) • C. Moments of the Sample • 1. Characteristics of Sample Mean

6. Sampling (cont.) • Example • Draw a single observation

7. Sampling (cont.) • Draw two observations

8. Sampling (cont.) • Draw 4 Observations

9. Sampling (cont.) • 2. Generalization • Every sample has an expected mean of . • But as our sample size increases, we are more confident of our results. • That is, the standard deviation (or standard error as we will call it) of our results is decreasing. • So as N increases,

10. Sampling (cont.) • 3. Hat Experiment • Mean = 10.5 • Standard deviation s = 5.77 • Now let's take a sample of size 1. (With replacement.) • Now one of size 2. • Now one of size 6.

11. Sampling (cont.) • 4. Equations • For a sample of size n from a population of mean and standard deviation , the sample mean has: • SE( ): it's called the standard error of the sampling process.

12. Inference We make inferences about a population from a given sample. • A. Population and Sampling Parameters • We have a population with parameters and . • We then take a sample with parameters and s. • We want to know how well the sample mean approximates the population mean .

13. Inference (cont.) • On average the sample mean equals the population mean.

14. Inference (cont.) • B. Referring Back to the Hat Experiment • 1. Sample Error decreases as n increases • For instance, before we drew samples of sizes 1, 2, and 6 from the hat. • The first sample of size 1 had standard error 5.77/ 1 = 5.77. • The second sample of size 2 had standard error 5.77/ 2 = 4.08. • The third sample of size 6 had standard error 5.77/ 6 = 2.36.

15. Inference (cont.) • C. Shape of the Sampling Distribution • If you take a sample and find its mean, then take another sample and find its mean and repeat this process a large number of times then • is a random variable with its own mean and standard error.

16. Inference (cont.) • 1. Central Limit Theorem • Take a large number of samples, then, the sample mean is normally distributed with mean and standard error .

17. Inference (cont.) • 2. Example: 3 different distributions • Example 1; • A population of men on a small, Eastern campus has a mean height =69" and a standard deviation =3.22". If a random sample of n=10 men is drawn, what is the chance that the sample mean will be within 2" of the population mean?

18. Inference (cont.) • Answer: • From the Central Limit Theorem, we know that is normally distributed, with mean 69 and standard error:

19. Inference (cont.) • Answer (cont.) • Find z-score • P(Z>1.96) = 0.025. Since there are two tails, the area in the middle is: So there's a 95% probability that the sample mean falls between 67 and 71.

20. Inference (cont.) • Example 2: • Suppose a large class in statistics has marks normally distributed around m = 72 with s = 9. Find the probability that • a) An individual student drawn at random will have a mark over 80.

21. Inference (cont.) • Answer: • The Z-score is (80-72)/9 = .89 • Looking this up in the table gives P(Z>.89) = .187, or about 19%. • b) Now, what's the probability that a sample of size 10 has an average of over 80?

22. Inference (cont.) • Answer: • The standard error is = 9/ 10 = 2.85. • So the Z-Score becomes (80-72)/2.85 = 2.81. • P(Z> 2.81) = .002.

23. Inference (cont.) • Example 3: I • f the number of miles per gallon achieved by all cars of a particular model has m = 25 and s = 2, what is the probability that for a random sample of 20 such cars, average miles per gallon will be less than 24? (assume that the population is normally distributed.) • Step 1: Standardize X

24. Inference (cont.) • Step 2: Then Find the Z scores (From the standard Normal tables) • So there is about a 1.3 percent chance that from a sample of 20 the average will be less than 24.

25. Inference (cont.) • D. Proportions • 1. Proportions as Means • A proportion (P) is just the mean of a dichotomous variable. • Example • Ask 50 people what they think of Clinton; • 0 if think he's doing a poor job; and • 1 if think he is doing a good job. • Suppose 30 of the 50 respondents say he's doing a good job • Then, the sample mean P is 30/50 = .60. • This is just another way of saying that 60% of those surveyed approved of his job performance.

26. Inference (cont.) • 2. Formula for Standard Error • For a large enough sample of size n, P (the proportion) will be normally distributed with mean and standard deviation . • Population Mean  = Population Proportion  • Sample Mean = Sample Proportion P • Population SD  =

27. Inference (cont.) • 3. Example: Polling • Suppose that the true approval rating for Clinton is .50. That is, 50 percent of the population believe he is doing a good job.  = .5 • If we sample 50 people, what is the probability that we will observe an approval rating as high as 60 percent or above?

28. Inference (cont.) • We know that the true population mean is =.5, • The Standard Error = = 0.0707 • Then the Z-score is (.6-.5) / 0.0707= 1.414 • Looking this up in the Z-table, P(Z>1.414) = .079, or about 8 %.

29. Inference (cont.) • 4.Example • Of your first 15 grandchildren, what is the chance that there will be more than 10 boys?

30. Inference (cont.) • Answer: • What the probability is that the proportion of boys is at least 10/15=2/3. • We know that the population mean is =1/2, • The standard error = • Then the Z-score is (.667-.5) / 0.129 = 1.29. • Looking this up in the table, P(Z>1.29) = .099, or about 10%.

31. Point Estimation: Properties • A. Unbiased Estimators • When an estimator has the property that it converges to the correct value, we say that it is unbiased.

32. Point Est. Properties (cont.) • B. Efficient Estimators • Def of Efficient: One estimator is more efficient than another if its standard error is lower.

33. Point Est. Properties (cont.) • C. N-1 Problem • 1. Known • When we take a sample of size n, if we had the real from the population, we could calculate • Then there wouldn't be a problem; would be a consistent estimator of , if we knew .

34. Point Est. Properties (cont.) • 2. Unknown • But we usually don't have , so we have to use the sample mean instead. What's the difference? Why don't we just say that • It turns out that we can show that minimizes the expression .

35. Point Est. Properties (cont.) • 2. Unknown (cont.) • So if we used instead, then, the expression would be bigger. • The right way to correct for this is to multiply by , so • The bottom line is that we use n-1 to make a consistent, unbiased estimate of the population variance.

36. IV. Review Homework • IV. Review Homework