1. Review Confidence Intervals Sample Size

2. Estimator and Point Estimate An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter. A Point Estimate is a single value or point used to approximate a population parameter. A point estimator may be biased or unbiased.

3. Central Limit Theorem Take ANY random variable X and compute  and  for this variable. If samples of size n are randomly selected from the population, then: 1) For large n, the distribution of the sample means, will be approximately a normal distribution, 2) The mean of the sample means will be the population mean  and 3) The standard deviation of the sample means will be

4. Confidence Intervals The Confidence Interval is expressed as: E is called the margin of error. For samples of size > 30,

5. Sample Size The sample size needed to estimate  so as to be (1-  )*100 % confident that the sample mean does not differ from  more than E is: …round up

6. Practice Problems #7.11 page 329 #7.19 page 331 #7.21 page 331

7. Small Samples What happens if and n is small (n < 30)? Our formulas from the last section no longer apply.

8. Small Samples What happens if and n is small (n < 30)? Our formulas from the last section no longer apply. There are two main issues that arise for small samples: 1)  no longer can be approximated by s 2) The CLT no longer holds. That is the distribution of the sampling means is not necessarily normal.

9. t- distribution If we have a small sample (n < 30) and wish to construct a confidence interval for the mean we can use a t-statistic, provided the sample is drawn from a normally distributed population.

10. t-distributions  is unknown so we use s (the sample standard deviation) as a point estimate of  We convert the nonstandard t-distributed problem to a standard t-distributed problem through the use of the standard t-score

11. t-distributions Mean 0 Symmetric and bell-shaped Shape depends upon the degrees of freedom, which is one less than the sample size. df = n-1 Lower in center, higher tails than normal. See Table inside front cover in text

12. Example In n=15 and after some calculation  2=0.025, t 0.025 = 2.145

13. Confidence Interval for the mean when  is unknown and n is small The (1-  )*100% confidence interval for the population mean  is The margin of error E, is in this case N.B. The sample is assumed to be drawn from a normal population.

14. Confidence Intervals for a small sample population mean The Confidence Interval is expressed as: The degrees of freedom is n-1.

15. Example The following are the heat producing capabilities of coal from a particular mine (in millions of calories per ton) 8,5008,3308,4807,960 8,030 Construct a 99% confidence interval for the true mean heat capacity. Solution: sample mean is 8260.0 sample Std. Dev. is 251.9 degrees of freedom = 4  = 0.01 7741.4    8778.6

16. Confidence intervals for a population proportion The objective of many surveys is to determine the proportion, p, of the population that possess a particular attribute. If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion.

17. Confidence intervals for a population proportion If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion. The idea here is to take a sample of size n, and count how many items in the sample have this attribute, call it x. Calculate the sample proportion,. We would like to use the sample proportion as an estimate for the population proportion.

18. Therefore at the (1-a)*100 % level of confidence, the Error estimate of the population proportion is At the (1-a)*100 % level of confidence, the confidence interval for the population proportion is :

19. Determining Sample Size In calculating the confidence interval for the population proportion we used Perhaps we might be interested in knowing how large a sample we should use if we are willing to accept a margin of error E with a degree of confidence of 1- .

20. Determining Sample Size If we already have an idea of the proportion (either through a pilot study, or previous results) one can use If we have no idea of what the proportion is then we use

21. In Class Exercises #7.31, 7.36, 7.41 Pages 341, 342 #7.50, 7.57 on page 349, 350 #7.75, 7.78 Pages 356, 357

22. Shortcut for finding z  Recall that as n  the Student’s T- distribution approaches the normal distribution. Look at T-table inside front cover, the last row represents the values of t n  as n becomes large which is essentially  z  Therefore, for some common values of  we are able to find z  quite quickly. z   z  

23. 23 Homework Review Chapter 7.3-7.5 Read Chapters 8.1-8.3 Quiz on Tuesday: Chapter 5