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Confidence Interval Estimation CQMS202 Sections 410 & 420 September 23, 2010 1 Outline Housekeeping items Review Confidence interval estimation for the mean (σ known) Confidence interval estimation for the mean (σ unknown) Confidence interval estimation for the proportion Determining sample size Confidence interval estimation and ethical issues 2 Housekeeping No enrollment Quiz 1 – October 7 – 45-60 mins Receive 75% refund if drop before October 7 3 Review Properties of normal distribution Transforming an observed value to standardized score 4 Problem 6.12 (p. 286, using calculator) Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1), what is the probability that: a. Z is less than 1.57? b. Z is greater than 1.84? c. Z is between 1.57 and 1.84? d. Z is less than 1.57 or greater than 1.84? 5 Sampling and sampling distribution Sample is used to estimate population characteristics Efficient, economical, more practical (but involves error) 6 Population vs. sample N n X i X i i1 X i1 N n ( X ) 2 ( X ) 2 X2 N s X2 n 1 N n 1 X X X Z Z X x x n 7 Population vs. Sample Population Parameters Sample Statistics - µ (mu) mean - (x-bar) mean M used in other places - σ (sigma) standard - s standard deviation deviation - σ2 (sigma-square) - s2 (s-square) variance variance - σx-bar (sigma-x-bar) - sx-bar(s-x-bar) standard error of the mean sample standard error 8 Problem 7.72 The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be a. Between 1.99 and 2.0 liters? b. Below 1.98 liters? c. Greater than 2.01 liters? d. The probability is 99% that the sample mean amount of soft drink will be at least how much? e. The probability is 99% that the sample mean amount of soft drink will be between which two values (symmetrically distributed around the mean)? 9 Confidence interval estimation for the mean (σ known) Deductive reasoning: population sample Inductive reasoning: sample population Estimation Point estimate: value of a single sample statistic (e.g., mean) Confidence interval estimate: numbers constructed around the point estimate 10 Estimating μ μ is rarely known. But we can estimate μ from sample means. There are some exceptions: estimated μ for IQ is around 100 with σ around 15. Standardization is achieved through a large number of testing in order to provide a more “accurate” estimation of the population μ. X X XU (z)( x ) (z)( / n ) z X X L (z)( x ) (z)( / n ) X (z)( / n ) 11 Confidence intervals in a graph μ=2 Sample 1 Sample 2 Sample 3 . . . . . Sample 100 12 Why z = ±1.96? With alpha (α) = .05, two-tailed, zcrit = ± 1.96 -z-crit +z-crit .95 .025 .025 13 Standard error, standard deviation & sample size Sampling error: n , σ-xbar x n Level of confidence: (1-α) x 100% X (z / 2 )( X (z / 2 )( ) ) n n z / 2 1.96, .05(95%confidence) z / 2 2.58, .01(99%confidence) 14 Caveats Assumption of normality Large sample size How to check? Data screening, especially data visualization skills such as stem-and-leaf plot and boxplot σ is not always known… 15 Problem 9.2 (p. 365) If X-bar = 125, σ = 24 and n = 36, construct a 99% confidence interval estimate of the population mean, μ. 16 Confidence interval estimation for the mean (σ unknown) t distribution William Gosset, 1908 Guinness Brewing Company Used the pseudonym of Student because the company would not allow him to publish under his name Develop the t distribution which relies on s2 instead of σ2. Sometimes called Student’s t distribution 17 Introduction to t statistics Working with what’s available z tests often require information about the population that is not available Population mean (μ) can be inferred from sample mean (M). But hard to decide the variance (hence standard deviation) of the population. Without σ, we cannot obtain std. err. (σM) for the z-formula. But t accommodates this limitation. 18 The t-distribution df = a + 2c df = a + c df = a 19 Comparing t to z distributions df = a + 2c df = a + c df = a z a=n–1 http://www.econtools.com/jevons/java/Graphics2D/tDist.html 20 Why did the t distribution fluctuates? Not enough info on the population can’t determine σ But we have the sample standard deviation: s ( X) 2 Calculating σ: X2 N N ( X ) 2 Calculating s: X2 n 1 s n 1 21 21 Degrees of Freedom Sample statistics are estimates of population parameters. n is a biased estimate of N the larger the n, the closer it is to N To obtain an unbiased estimator, some adjustment is necessary. Degrees of freedom (df) indicates the number of obs in a sample minus the number of estimated parameters. For calculating s, the estimated parameter is σ. Hence, n – 1 22 Standard Error for one sample It is more often that we deal with a sample than an individual. To estimate a population from a sample, we need sM instead of s. 2 s s sM n n We’ll come back to the s.e. again… 23 Confidence interval estimation for the mean (σ unknown) X (z / 2 )( ) X (z / 2 )( ) n n s s X (t / 2 )( ) X (t / 2 )( ) n n df for t = n - 1 24 Problem 9.17 (p.373) The data below represent the total fat, in grams per serving, for a sample of 20 chicken sandwiches from fast-food chains. 7, 8, 4, 5, 16, 20, 20, 24, 19, 30, 23, 30, 25, 19, 29, 29, 30, 30, 40, 56 a. Construct a 95% confidence interval for the population mean total fat, in grams per serving. b. Interpret the interval constructed in a. c. What assumption must you make about the population distribution in order to construct the confidence interval estimate in a? d. Do you think that the assumption needed in order to construct the confidence interval estimate in a is valid? Explain. 25 Confidence interval estimation for the proportion Population proportion: π Point estimate for π is the sample proportion: p = X/n p(1 p) p(1 p) p (z / 2 )( ) p (z / 2 )( ) n n z / 2 1.96, .05(95%confidence) z / 2 2.58, .01(99%confidence) X p ~ X,(n X) 5 n 26 Problem 9.27 (p. 377) The start of the twenty-first century saw many corporate scandals and many individuals lost faith in business. In a 2007 poll conducted by the NYC-based Edelman Public Relations firm, 57% of respondents say they trust business to “do what is right”. This percentage was the highest in the annual survey since 2001. a. Construct a 95% confidence interval estimate of the population proportion of individuals who trust business to “do what is right” assuming that the poll surveyed: 1. 100 individuals 2. 200 individuals 3. 300 individuals b. Discuss the effect that sample size has on the width of confidence intervals. c. Interpret the intervals in a. 27 Determining sample size e = sampling error X z / 2 n (1 ) e z / 2 e z / 2 n n Point estimate x-bar z / 2 n z / 2 (1 ) 2 2 2 n 2 2 e e 28 Problem 9.39 (p. 382) An advertising agency that serves a major radio station wants to estimate the mean amount of time that the station’s audience spends listening to the radio daily. From past studies, the stand deviation is estimated as 45 minutes. a. What sample size is needed if the agency wants to be 90% confident of being correct to within ±5 minutes? b. If 99% confidence is desired, how many listeners need to be selected? 29 Problem 9.43 (p. 383) A study of 658 CEOs conducted by the Conference Board reported that 250 stated that their company’s greatest concern was sustained and steady top-line growth. a. Construct a 95% confidence interval for the proportion of CEOs whose greatest concern was sustained and steady top- line growth. b. Interpret the interval constructed in a. c. To conduct a follow-up study to estimate the population proportion of CEOs whose greatest concern was sustained and steady top-line growth to within ±.01 with 95% confidence, how many CEOs would you survey? 30 Review problems 9.52, 9.54, 9.58 Next week Chapter 10 31

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