1. The Practice of Statistics Third Edition Chapter 10: Estimating with Confidence Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

2. Ex. Suppose a sample of 50 men had a mean score of 109 on an intelligence test. We can estimate that the population mean , is approximately 109. x bar is normally distributed. The mean of the sampling distribution is equal to  the unknown population mean. The standard deviation of x bar for an SRS of 50 given the population standard deviation  is 15/(50) 0.5 = 2.1

3. The 68 – 95 – 99.7 rule states that about 95% of all possible sample means x bar will be within 2 standard deviations of the population mean 

4. In 95% of all possible samples the unknown , lies between x bar + or – 4.2 We are 95% confident that  lies between 109 + 4.2; that is (104.8, 113.2) There are only two possibilities: 1.The interval between 104.8 and 113.2 contains the true population mean  2.Our SRS was one of the few samples for which x bar is not within 4.2 points of the true  Only 5% of all samples give such inaccurate results.  The method we used gives the correct result 95% of the time.

5. Applet showing confidence intervals: http://onlinestatbook.com/stat_sim/conf_interval/index.html

6. Suppose you want to construct an 80% confidence interval  Confidence level Tail areaZ*Z* 80%0.11.282 90%0.051.645 95%0.0251.960 99%0.0052.576  Confidence level is usually chosen as > 0.90

8. Margin of error estimate Ex. A questionnaire of 160 hotel managers asked how long they had been with their current company. The average time was reported as 11.78 years. Give a 99% confidence interval for the mean number of years that the entire population of managers have been with there current company. Assume the standard deviation of the population is   years. 11.78 + 2.576(3.2/√160) = 11.78 + 0.652 = (11.128, 12.432) We are 99% confident that the true population mean lies between 11.128 and 12.432. The method we used will give the correct result 99% of the time.

9. Margin of error decreases when; 1)z* gets smaller; but this makes confidence level smaller  is small – sample drawn from less spread population. 3)n, sample size is large. Quadrupling the sample size cuts margin of error in half. Ideally, we would like; 1) high confidence; method almost always gives the right result. and 2) small margin of error; population parameter estimated very precisely.

10. How to choose a sample size for a desired margin of error. Ex. How many observations must be made to produce results accurate to within + 0.005 with 95% confidence? Assume  z*  /√n    n => 7.1 < n ; choose n greater than or equal to 8 You must round up to next integer

11. It is incorrect to say that the probability is 95% that the true mean lies within a certain interval. We can say that we are 95% confident that the mean lies within a certain interval or ; The method we used to calculate the interval gives the correct result in 95% of all possible sample of a particular size.

12. Tests of Significance Significance tests assess the evidence provided by the data in favor of some claim about the population. Significance tests begin by stating a hypothesis about a population parameter. The null hypothesis H o, is always stated as an equivalence. H o :  o The alternative hypothesis H a, can be stated in one of three ways. H a :  ≠  o  <  o  >  o

13. Ex. A car manufacturer claims that one of their car models gets 33mpg. A random sample of 30 cars is selected and the mean gas mileage of this sample x-bar is calculated to be 31 mpg. Can we refute the claim of the automaker? Assume  3.5 mpg. Ho:  33 mpg Ha:  mpg x - bar = 31 mpg, sample std. = 3.5/√30 = 0.639

14. 33 3.5/√30 = 0.639 33 3132.361 - 0.639

15. 3331 33 0.00087 X-bar = 31 is way out on the normal curve. So far out that a result this small almost never occurs by chance if the true  33 mpg. This is good evidence that the automakers claim should be rejected in favor of the alternate hypothesis,  33 mpg Generally P-values < 0.05 are considered small enough to reject the Ho. It is statistically significant. P( z < -3.12) = 0.00087

17. Significance level We compare the P – value with a fixed value that we regard as decisive. The decisive value of P is called the significance level. Symbol =>  Choosing  = 0.05 require that the data give evidence against Ho so extreme that it would happen in no more than 5% of the possible samples if Ho is true.  = 0.01 require that the data give evidence against Ho so extreme that it would happen in no more than 1% of the possible samples if Ho is true. If the P – value is as small or smaller than , we say that the data are statistically significant at level  The null hypothesis should be rejected in favor of the alternate hypothesis. If P-value is low, reject the HO

18. One sided test Two sided test {

21. Choosing an  level in significance tests If H o represents an assumption that people you must convince have believed for a long period of time, strong evidence (small  is needed to persuade them. If the consequences of rejecting H o are drastic; ie expensive, finality. You may want strong evidence, (small  ). May be more useful to report the P-value so each individual may decide for themselves. Even though significance levels of 0.10, 0.05 and 0.01 have been used traditionally. The border between what levels are significant is not black and white. Not much difference between P-values of 0.049 and 0.051. No significance level is sacred.

22. Inference as decision Type I and Type II errors If we reject Ho (accept Ha) when Ho is really true, this is a Type I error. If we reject Ha (accept Ho) when Ha is really true, this is Type II error. H o TrueH a True Reject H o Type I Error Correct Decision Reject H a Correct Decision Type II error

23. Significance and Type I error The significance level  of any fixed level significance test is equal to the probability of making a Type I error. the value of  is the probability that the test will reject the null hypothesis Ho when Ho is really true. Power of the test The probability that a fixed level a significance test will reject Ho when Ha is true is called the power of the test. Increasing sample size n, increases the power of the test. Increasing the significance level  increases the power of the test.