Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Topics for Today More on Confidence Intervals Stat203 Fall 2011 – Week 6 Lecture 3 Page 1 of 28 Example NAEP quantitative scores The NEAP is a broad-scope survey conducted on a _____________ of students in grade 4, 8 and 12 in the United States. http://en.wikipedia.org/wiki/National_Assessment_of_Educational_Progress It’s used by law makers and educators to form policy and priorities. Such as comparing states: [ source http://www.schoolinfosystem.org/archives/naep_state.gif ] Stat203 Fall 2011 – Week 6 Lecture 3 Page 2 of 28 and tracking the reading performance of students (scores are from the ‘reading’ component of the survey) [ source: http://www.balancedreading.com/2005NAEP.gif ] Stat203 Fall 2011 – Week 6 Lecture 3 Page 3 of 28 … but there are many components: Stat203 Fall 2011 – Week 6 Lecture 3 Page 4 of 28 The NAEP mathematics section test basic arithmetic skills and gives a ‘quantitative score’. Scores on the test range from 0 to ___. A person who scores 233 can add the amounts of two cheques appearing on a bank deposit slip. Someone scoring 325 can determine the price of a meal from a menu. A person scoring ___ can transform a price in cents per ounce into dollars per pound. Stat203 Fall 2011 – Week 6 Lecture 3 Page 5 of 28 In a recent year, ___ U.S. men 21 to 25 years of age were in the NAEP sample. Their mean quantitative score was x = ___. These ___ men are a _____________ from the population of all young men. Stat203 Fall 2011 – Week 6 Lecture 3 Page 6 of 28 On the basis of this sample, what can we infer about the mean score μ in the population of all 9.5 million young men of these ages in the U.S.? Stat203 Fall 2011 – Week 6 Lecture 3 Page 7 of 28 The ____________________ tells us the sample mean x from a random sample will be close to the population mean μ. x = 272 so μ is somewhere around 272 But how _____? To determine how close it is, we need to remember the _____________________ of x , and use it to construct a confidence interval. Stat203 Fall 2011 – Week 6 Lecture 3 Page 8 of 28 Sampling distribution of x : _____________________: the mean x of 840 scores has a distribution that is approximately ______. The mean of this normal sampling distribution is the same as the mean, μ of the entire population. The __________________ of x from a sample of 840 is 840 , where σ is the standard deviation of the distribution of individual scores. Stat203 Fall 2011 – Week 6 Lecture 3 Page 9 of 28 Suppose σ = __. The standard deviation of x is 60840 = ___ If we choose many samples of size 840 and find the means and display their distribution, it would look more like a ______ distribution with mean μ and standard deviation = 2.1. Stat203 Fall 2011 – Week 6 Lecture 3 Page 10 of 28 Statistical Confidence The 68-95-99.7 rule says that in 95% of all samples, the mean x will be within _ standard deviations of the _______________ score μ. So in 95% of the _______ the population parameter μ is between x - 4.2 and x + 4.2 Our sample gave x = ___ Stat203 Fall 2011 – Week 6 Lecture 3 Page 11 of 28 Statistically speaking we are 95% confident that the unknown mean (μ) score lies between x - 4.2 = 272 - 4.2 = ______ and x + 4.2 = 272 + 4.2 = _____ 95% confidence implies the method gives correct results 95% of the time. x 4.2 ________________ is called a 95% confidence interval for μ. Stat203 Fall 2011 – Week 6 Lecture 3 Page 12 of 28 Recognize there are 2 possibilities: 1. The interval between 267.8 and 276.2 ________ the true population parameter μ. 2. Our sample was one of the few samples for which x is __________ 4.2 points of μ. Only __ of the samples will give such results. [ source http://www.southalabama.edu/coe/bset/johnson/lectures/lec16_files/image006.jpg ] Stat203 Fall 2011 – Week 6 Lecture 3 Page 13 of 28 Stat203 Fall 2011 – Week 6 Lecture 3 Page 14 of 28 Confidence Interval Estimate ± _______________ Estimate: Our guess for the unknown parameter. The estimate in our case is the statistic x . Margin of Error: Measures the _________ of our estimate, based on the variability of the estimate. Confidence Level: ___________ that the interval will _______ the true parameter. In repeated sampling we would expect 95% of the intervals to contain the true population parameter μ. Stat203 Fall 2011 – Week 6 Lecture 3 Page 15 of 28 Margin of Error A __% confidence interval for the population mean μ has a margin of error ____ * (standard deviation of x ) = 1.96 * n The margin of error for a __% confidence interval for the mean μ is given by: _____* n The margin of error for a __% confidence interval for the mean μ is given by: ____ * n Stat203 Fall 2011 – Week 6 Lecture 3 Page 16 of 28 Example: Interpretting a CI A poll on voting preferences for candidates interviewed 1025 people randomly selected in the Vancouver area. The poll found that 46 % of the people surveyed said they preferred the Liberal party. a) The poll announced a margin of error of ± 3 percentage points with a 95% confidence level. What is the 95% confidence interval for the percent of all people who will vote Liberal in the upcoming election? b) What are your conclusions? Stat203 Fall 2011 – Week 6 Lecture 3 Page 17 of 28 Example: Analyzing chemical data A manufacturer of chemical products analyses a sample from each batch of a product to verify the concentration of a particular ingredient. The chemical analysis is not very accurate. Repeated measurements on the same batch give different results and are approximately normally distributed. The analysis procedure has no bias and the population mean μ is the true concentration of the sample. The standard deviation of this distribution is known to be σ = 0.0068 grams/litre. The lab analysed each sample three times and reported the average reading. Stat203 Fall 2011 – Week 6 Lecture 3 Page 18 of 28 Three analyses of one sample give the following concentrations: 0.8403 0.8363 0.8447 a) Construct a 95% confidence interval for the true concentration μ. Stat203 Fall 2011 – Week 6 Lecture 3 Page 19 of 28 b) Management asks the lab to produce results that are accurate to within ± 0.001 with a 95% confidence. How many samples should be taken to comply with this request? Stat203 Fall 2011 – Week 6 Lecture 3 Page 20 of 28 Stat203 Fall 2011 – Week 6 Lecture 3 Page 21 of 28 In reality, is usually UNKNOWN Up to now, all the examples have given you the standard deviation from the __________ (), but this is rarely (if ever) known. Instead, we will have to use the standard deviation from the ______ (s) Stat203 Fall 2011 – Week 6 Lecture 3 Page 22 of 28 Remember, the standard deviation and s mean the same thing, just one is measured on the __________ and the other is measured on the ______. However, when the population standard deviation is unknown, we ____ to use the sample standard deviation. This doesn’t come for free. Since we’re estimating something else from the sample (instead of just ‘knowing’ it), our confidence intervals get _____. Stat203 Fall 2011 – Week 6 Lecture 3 Page 23 of 28 The t-distribution The t-distribution is a similar shape to the normal distribution. (t in red, normal in blue) [source http://www.mechanical-writings.com/img/gt/confidence-interval-t-distribution/T_distribution_1df.png ] Stat203 Fall 2011 – Week 6 Lecture 3 Page 24 of 28 Unlike the normal distribution, (which depends only on µ and ), the t-distribution depends on something called the degrees of freedom. [source http://02.edu-cdn.com/files/static/mcgrawhillprof/9780071621885/ESTIMATION_AND_CONFIDENCE_INTERVALS_07.GIF ] Stat203 Fall 2011 – Week 6 Lecture 3 Page 25 of 28 or … you can check out this applet: http://www.stat.tamu.edu/~jhardin/applets/signed/T.html look at the panel on the left side and change the degrees of freedom (slider at the top of the page) to see how the t-distribution becomes more ‘normal’ as the degrees of freedom increase. Our margin of error will use the t-distribution instead of the z. More on this … next time. Stat203 Fall 2011 – Week 6 Lecture 3 Page 26 of 28 Today’s Topics Confidence Intervals - Constructing and interpreting confidence intervals - t-statistic is used if is unknown Stat203 Fall 2011 – Week 6 Lecture 3 Page 27 of 28 Reading for next lecture No New Reading Stat203 Fall 2011 – Week 6 Lecture 3 Page 28 of 28