1. Chapter 7, part D

2. VII. Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample proportion. A. Expected Value E( ) = p E( ) =.60 in the EAI problem.

3. B. Standard Deviation For a finite population: For an infinite population: If n/N .05 it’s acceptable to consider the population an infinite one.

4. EAI Example With a sample size of n=30, n/N=.012 so we can calculate the standard error using the infinite equation:

5. C. Form of the Sampling Distribution of You can apply the Central Limit Theorem: The sampling distribution of can be approximated by a normal probability distribution when n is large. How large is “large”? If np  5 and n(1-p)  5 np=30(.6)=18 n(1-p)=30(.4)=12 so we can use the CLT.

6. D. Practical Value Again, we use the CLT and our sampling distribution to estimate probabilities that a sample proportion is fairly close to the true population proportion. Suppose our personnel director wants an estimate of that is within.05 of the true p.

7. The Problem What is the.55.60.65 Find this area and multiply by 2.

8. The Solution Find the z-score: z = (.65-.60)/.0894 =.559 or.56 From the standard normal table, the area for z=.56 is.2123. Thus or the manager’s desired level of accuracy will happen less than 43% of the time. Redo this problem with n=100 to see how accuracy improves.

9. VIII. Properties of Point Estimators Before assuming that a point estimator is an appropriate estimator of the population parameter, we must check 3 criteria. Let’s say  is a population parameter and (theta- hat) is a sample statistic or point estimator of .

10. A. Unbiasedness is an unbiased estimator of  if E( ) = . E( )=  This is an unbiased estimator. Sampling errors above and below  balance each other out.

11. Biased Estimator If our estimator is biased like below, sampling errors above  are very probable. Thus the sample statistic has a very high probability of overestimating .  E( ) In later chapters we will take greater pains to ensure that we have unbiased estimators.

12. B. Efficiency Given two sampling distributions of equal mean (  ) we would prefer the one with the smallest standard deviation. The point estimator with the smallest s is said to have greater relative efficiency than the other. Why? Because probabilities (and accuracy) increase as the standard deviation falls. We’ve seen this in the EAI problem.

13. C. Consistency is consistent if values of tend to become closer and closer to  as n . To see this, look at how increasing n provides point estimates of that are closer to . As n , thus the distribution becomes more narrow and the values of gather closer to .