STA 291 - Lecture 20 1STA 291Lecture 20• Exam II Today 5-7pm• Memorial Hall (Same place as exam I)• Makeup Exam 7:15pm – 9:15pm• Location CB 234• Bring a calculator, picture IDExam II Covers …• Chapter 9• 10.1; 10.2; 10.3; 10.4; 10.6• 12.1; 12.2; 12.3; 12.4• Formula sheet; normal table; t-table will be provided. STA 291 - Lecture 20 2Example:• Smokers try to quit smoking with Nicotine Patch or Zyban.• Find the 95% confidence intervals STA 291 - Lecture 20 3Example• To test a new, high-tech swimming gear, a swimmer is asked to swim twice a day, one with the new gear, one with the old.• The difference in time is recorded: Time(new) – time(old) = -0.08, -0.1, 0.02, …. -0.004. There were a total of 21 such differences.Q: is there a difference?STA 291 - Lecture 20 4• First: we recognize this is a problem with mean mu.• And we compute the average X bar= -0.07• SD = 0.02• 90% confidence interval is:STA 291 - Lecture 20 5Plug-in the values into formulaSTA 291 - Lecture 20 60.020.02?? and ??2121XX−+0.020.020.07?? and 0.07??2121−−−+• What is the ?? Value.• It would be 1.645 if we knew sigma, the population SD. But we do not, we only know the sample SD. So we need T-adjustment.• Df= 21 -1 = 20• ??=1.725STA 291 - Lecture 20 7Confidence interval for mu• For continuous type data, often the parameter is the population mean, mu.• Chap. 12.1 – 12.4STA 291 - Lecture 20 8STA 291 - Lecture 20 9Chap. 12.1 – 12.4:Confidence Interval for mu• The random interval betweenWill capture the population mean, mu, with 95% probability1.96 and 1.96XXnnσσ−+• This is a confidence statement, and the interval is called a 95% confidence interval• We need to know sigma. LSTA 291 - Lecture 20 10• confidence level 0.90, çè =1.645• confidence level 0.95 çè =1.96• confidence level 0.99 çè =2.575• Where do these numbers come from? (same number as the confidence interval for p). • They are from normal table/web/2zα/2zα/2zαSTA 291 - Lecture 20 11“Student” t - adjustment• If sigma is unknown, (often the case) we may replace it by s (the sample SD) but the value Z (for example z=1.96) needs adjustment to take into account of extra variability introduced by s• There is another table to look up: t-table or another applet• http://www.socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htmDegrees of freedom, n-1• Student t - table is keyed by the df –degrees of freedom• Entries with infinite degrees of freedom is same as Normal table• When degrees of freedom is over 200, the difference to normal is very smallSTA 291 - Lecture 20 12• With the t-adjustment, we do not require a large sample size n.• Sample size n can be 25, 18 or 100 etc.STA 291 - Lecture 20 13STA 291 - Lecture 20 14STA 291 - Lecture 20 15Example: Confidence Interval• Example: Find and interpret the 95% confidence interval for the population mean, if the sample mean is 70 and the pop. standard deviation is 12, based on a sample of size n = 100First we compute =12/10= 1.2 , 1.96x 1.2=2.352[ 70 – 2.352, 70 + 2.352 ] = [ 67.648, 72.352] nσSTA 291 - Lecture 20 16Example: Confidence Interval• Now suppose the pop. standard deviation is unknown (often the case). Based on a sample of size n = 100 , Suppose we also compute the s = 12.6 (in addition to sample mean = 70)First we compute =12.6/10= 1.26 , From t-table 1.984 x 1.26 = 2.4998[ 70 – 2.4998, 70 + 2.4998 ] = [ 67.5002, 72.4998] snSTA 291 - Lecture 20 17Confidence Interval: Interpretation• “Probability” means that “in the long run, 95% of these intervals would contain the parameter”i.e. If we repeatedly took random samples using the same method, then, in the long run, in 95% of the cases, the confidence interval will cover the true unknown parameter• For one given sample, we do not know whether the confidence interval covers the true parameter or not. (unless you know the parameter)• The 95% probability only refers to the method that we use, but not to the individual sampleSTA 291 - Lecture 20 18Confidence Interval: Interpretation• To avoid the misleading word “probability”, we say:“We are 95% confident that the interval will contain the true population mean”• Wrong statement: “With 95% probability, the population mean is in the interval from 3.5 to 5.2”Wrong statement: “95% of all the future observations will fall within 3.5 to 5.2”.STA 291 - Lecture 20 19Confidence Interval• If we change the confidence level from 0.95 to 0.99, the confidence interval changesIncreasing the probability that the interval contains the true parameter requires increasing the length of the interval• In order to achieve 100% probability to cover the true parameter, we would have to increase the length of the interval to infinite -- that would not be informative, not useful.• There is a tradeoff between length of confidence interval and coverage probability. Ideally, we want short length and high coverage probability (high confidence level).STA 291 - Lecture 20 20Different Confidence Coefficients• In general, a confidence interval for the mean, has the form• Where z is chosen such that the probability under a normal curve within z standard deviations equals the confidence levelµXznσ±⋅STA 291 - Lecture 20 21Different Confidence Coefficients• We can use normal Table to construct confidence intervals for other confidence levels• For example, there is 99% probability of a normal distribution within 2.575 standard deviations of the mean • A 99% confidence interval for isµ2.575Xnσ±⋅STA 291 - Lecture 20 22Error Probability• The error probability (a) is the probability that a confidence interval does not contain the population parameter -- (missing the target)• For a 95% confidence interval, the error probability a=0.05• a = 1 - confidence level orconfidence level = 1 – aSTA 291 - Lecture 20 23Different Confidence LevelsConfidence levelError a a/2 z90% 0.195% 0.05 0.025 1.9698% 0.02 0.01 2.3399% 2.57599.74% 386.64% 0.1336 0.0668 1.5• If a 95% confidence interval for the population mean, turns out to be[ 67.4, 73.6]What will be the confidence level of the interval [ 67.8, 73.2]?STA 291 - Lecture 20 24Choice of sample size• In order to achieve a margin of error smaller than B, (with confidence level 95%), how large the sample size n must we get?STA