Math 113 Section 21-26: Margin of Error and Confidence Intervals - Prof. Mingqing Xiao, Study notes of Mathematics

Download Math 113 Section 21-26: Margin of Error and Confidence Intervals - Prof. Mingqing Xiao and more Study notes Mathematics in PDF only on Docsity! 1 Announcement Math 113 section 21-26 Instructor: M.Xiao •The Third Hour Exam: Nov. 6, Friday in class. Oct. 30, 2009 Help sessions Tuesday and Thursday 6:30-7:45p.m. Location: agriculture building (AG) 152. Course web page: http://kalman.math.siu.edu/~mxiao/math113.html Margin of Error The margin of error announced by most national samples says how close to the truth about the population the sample result would fall in 95% of all samples drawn by the method used to draw this one sample We say that we have 95% confidence that the truth about the population lies within the margin of error. Example “A new poll shows that 60% of all Americans approve of the way the president is handling his office. The margin of error for the poll is plus or minus 3%” the true percentage is almost certain (95%) to be within the 57%-63% range. It means: The Quick Method for Margin of Error Use the sample percent from a simple random sample of size n to estimate an unknown population percent. The margin of error for 95% confidence is roughly equal to 100/ n Old Exam Question A poll asked 956 licensed drivers whether they supported a nationwide lowering of the drunk driving limit to 0.08% BAC (blood alcohol content), and 72% said they did. Use the quick method to estimate the margin of error for conclusion about all Licensed drivers. A) 30.9% B) 3.2% C) 32.3% D) 11.8% 23.3 919.30 100 956 100100 === n Sample Proportion Question: what is the percentage of Americans approve of the way the president is handling his office? True proportion = The number of Americans who say yes Total of all Americans Sample proportion = The number of samples who say yes sample size pIn our textbook stands for true proportion. p̂ stands for sample proportion. 2 An Example Suppose you conduct a telephone poll of 1250 people, asking them whether or not they favor mandatory sentencing for drug related crimes. If 580 people say “yes,” what is the sample proportion of people in favor of mandatory sentencing? p̂ Answer: %4.46464.1250 580ˆ ===p Old Exam Question A random sample of 300 car owners in Louisville indicated that 36 had full insurance coverage on glass breakage for their car, with no deductible amount. In this example, What is the sample proportion? __A. 83% __B. 36% __C. 8.3% __D. 12% %1212.0 300 36 ==Answer: Standard Deviation of Sample Proportion For a given sample size n, the sample proportion •varies with the sample •is closer to a normal distribution when n is large •the mean is the true proportion •the standard deviation of the sampling distribution is p n pp p )1( ˆ − =σ Example Suppose you conduct a telephone poll of 1250, asking them whether or not they favor mandatory sentencing for drug related crimes. If the true population proportion is p=45%, what is the standard deviation of the sampling distribution? Answer: %41.1 0140712472.0000198.0 1250 )45.01(45.0)1( ˆ = ==−=−= n pp pσ Confidence Intervals Choose an SRS of size n from a large population that contains an unknown percent p of successes. A 95% confidence interval for p is n pp p )̂1(̂ 2ˆ − ± Here is the proportion of successes in the sample. Both p and are measured in percent. p̂ p̂ Example: Risky Behavior in the Age of AIDS The National AIDS Behavior Surveys interviewed a random sample of 2673 adult heterosexuals. Of these, 170 had more than one sexual partner in the past year. The sample proportion who admit to multiple partners is %36.6 2673 170ˆ ==p A 95% confidence interval for the proportion p of all adult heterosexuals with multiple partners is therefore %42.5%94.0%36.6 2673 %)36.61%(36.62%36.6)̂1(̂2ˆ =±=−±=−± n ppp to %30.7