1 Chapter 9: Sampling Distributions

2 Activity 9A, pp

3 We’ve just begun a sampling distribution! Strictly speaking, a sampling distribution is: A theoretical distribution of the values of a statistic (in our case, the mean) in all possible samples of the same size (n=100 here) from the same population. Sampling Variability: The value of a statistic varies from sample-to- sample in repeated random sampling. We do not expect to get the same exact value for the statistic for each sample!

4 Sampling Distribution The sampling distribution answers the question, “What would happen if we repeated the sampling or experiment many times?” Formal statistical inference is based on the sampling distribution of statistics.

5 Definitions Parameter: A number that describes the population of interest. Rarely do we know its value, because we do not (normally) have all values of all individuals from a population. We use µ and σ for the mean and standard deviation of a population. Statistic: A number that describes a sample. We often use a statistic to estimate an unknown parameter. We use x-bar and s for the mean and standard deviation of a sample.

6 Problems , p. 489: Parameter or Statistic?

7 Example 9.4, p. 491 Compare Figures 9.2 and 9.4

8 Probability Distribution of Random Digits

9 All possible samples of size n=2

10 Sampling Distribution of the Mean

11 Exercise 9.7, p. 494

12 What happens to a sampling distribution when we increase our sample size (n)? Example 9.5, pp

13 Results of 1000 SRSs of size n=100

14 Results of 1000 SRSs of size n=1000

15 Expanded scale of previous slide

16 Statistic Bias If the mean of the sampling distribution is equal to the population parameter, the statistic is said to be unbiased. Now, be careful—the sample mean you actually get may in fact be “off” the parameter mean. However, there is no systematic tendency, on repeated samplings, to overestimate or underestimate the parameter.

17 Variability of Statistic (pp ) “Properly chosen statistics computed from random samples of sufficient size will have low bias and low variability.”

18 Figure 9.9, p. 500

19 Spread of a sampling distribution As long as N>10n, the spread of the sampling distribution does not depend on the size of the population. National poll (300,000,000): need approx. n=1,100 for ±3% margin of error. Asheville city poll (70,000): need approx. n=1,100 for ±3% margin of error. See p. 498 for discussion.

20 Homework Read through p , p and 9.17, p. 503

Sample Proportions We use p ^ as an estimate of p (the parameter). What does the sampling distribution of p ^ look like? Knowing the center, shape, and variability of the sampling distribution will give us an idea of how confident we can be in using p ^ as an estimate of p. If the population is at least 10X larger than the sample, we can use binomial distribution facts to develop equations for the mean and standard deviation of a sampling distribution for p ^ :

22 Sampling distribution for proportion

23 Using the Normal Approximation for p ^ Example 9.5 showed us that for large samples, the sampling distribution of p ^ is approximately normal (pp ). Following the convention of this text, we will use the normal approximation for the sampling distribution of p ^ as long as the following conditions are satisfied: Using the normal approximation is quite accurate if the above conditions are met, plus we can take advantage of the useful standard normal probability calculations.

24 Exercises Read over Example 9.7, p. 507 Be sure to read Example 9.8 tonight. Exercise 9.19, p. 511

25 Homework Problems: 9.22, p , p. 514 Reading through p. 514 Quiz, Wednesday Chapter 9 Test on Thursday

Sample Means In 9.2 we were dealing with a sample proportion. This statistic is used when we are interested in some categorical variable. In 9.3 we switch to looking at the sample mean. Used for quantitative variables.

27 Sampling Distribution for a Sample Mean See bulleted list on p. 516: Sample mean x-bar is an unbiased estimator of the population mean µ. The values of x-bar are less spread out for larger samples. Box on p. 517 The text tells us that if we draw a SRS of size n from a normal distribution, the sampling distribution will also be normal. But what about drawing samples from a population whose distribution is not normal?

28 The Central Limit Theorem (p. 521) One of the more important ideas of statistics. If we draw a sample that is large enough … …the sampling distribution is approximately normal no matter what the shape of the underlying distribution! How large the sample must be to get close to a normal distribution depends on the shape of the underlying distribution, but samples of size n=25 to n=30 generally suffice.

29 Example 9.12, p. 521 (exponential distribution)

30 Exercises 9.31, p , p. 524

31 Exercise 9.31 Important ideas: Averages are less variable than individual observations. Averages are more normal than individual observations.

32 Homework Exercises 9.39 through 9.42, pp Test on Thursday