- Decide on an estimator for the parameter. - Calculate distribution of estimator; usually involves unknown parameter

Transcription

1 General Confidence Intervals 1 - Decide on an estimator for the parameter - Calculate distribution of estimator; usually involves unknown parameter - Convert a confidence statement about where the estimate will fall into a confidence statement about the unknown parameter t Confidence Interval Example 2 Radiocarbon dates are given for a sample of 9 observations taken from an archaeological excavation of the Danebury iron-age hill fort. It is determined that the distribution of radiocarbon ages is symmetric and approximately normal. Find a 95% and a 99% C.I. for the mean age of the archaeological site. The sample average, x = years, the sample standard deviation, s = years, and d.f. = 9-1 = 8 1

2 t Confidence Interval Example 3 Chapter 7 Exercise 98 4 Data collected on random sample of 32 first-episode schizophrenia patients. Mean and standard deviation for number of years of education for patients in sample were 12.4 years and 3.0 years respectively. Construct a 95% confidence interval for mean number of years of education for all such patients. 2

3 Chapter 7 Exercise 98 5 Data collected on random sample of 32 first-episode schizophrenia patients. Mean and standard deviation for number of years of education for patients in sample were 12.4 years and 3.0 years respectively. Construct a 95% confidence interval for mean number of years of education for all such patients. Robustness of t approach 6 A statistical inference procedure is robust if the probability calculations are not sensitive to violations of the assumptions. We needed to assume that the sample average was normally distributed in order to carry out the t procedures. For small samples, n < 30 might worry about validity of this assumption. t procedure will work quite well as long as data contains no extreme outliers For large samples n 30, normality assumption not a problem. Also s will approach σ, and we could fairly safely use Z procedures (confidence intervals based upon normal distribution) 3

4 Inference for Proportions 7 Example: Nicotine replacement programs together with counseling generally have a success rate of about 20% in helping smokers quit. A random sample of 200 people who participate in a new nicotine gum/counseling program is selected. Fifty-five (55) of the 200 were successful in quitting smoking. What proportion of all smokers who use the new gum/counseling program will successfully quit smoking? Construct a 95% confidence interval for the true proportion of smokers who will successfully quit. Is there evidence to suggest that the new program is more effective than existing nicotine replacement programs together with counseling? Confidence intervals based upon normal distribution 8 Suppose we want to produce confidence intervals for a population proportion in the usual fashion: estimate ± margin of error Need an appropriate statistic (estimate): Need variability associated with the estimate: If we can assume that the sample proportion has approximately a normal distribution then: or using our approximation for p: Back on familiar ground. Confidence intervals for the population proportion p are: 4

5 Confidence interval for proportion of smokers who quit 9 In our example, pˆ = 27.5%, n=200, and our estimate of σ p ˆ =.275(.725) / 200 = So a 95% confidence interval for p is ±1.96(0.0316) = ±0.063 or 21.2% to 33.8% Our margin of error (error bound), E, is: What if we wished to ensure that our 95% C.I. had a margin of error no greater than 3%? Confidence Intervals for p - Sample Size Determination 10 If we want a specified margin of error, we will need to choose our sample size accordingly. Our margin of error, So we could just solve for n in each case to get our desired sample size, 5

6 Confidence Intervals for p - Sample Size Determination 11 Do not know pˆ until we have collected the sample. May have an idea of what value pˆ is likely to assume based upon earlier study, or we could imagine a worst case scenario, the largest sample we might need, which occurs when If we determine a sample size using this assumption, then This will give us a sample size at least large enough to achieve our desired margin of error and will usually do better. Self-Testing Review 7.6: Exercise Preliminary sample of 512 employees questioned to assess prevalence of symptoms attributed to work environment. Forty-five(45) employees reported experiencing eye irritation. How large a random sample needed to be 90% confident of being within.03 of the population proportion? 6