Slide Number 1Slide Number 2Slide Number 3We want to ensureSlide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Critical values for confidence intervals for a mean (known σ)Slide Number 15Example: 95% confidence interval for a mean (known σ)Slide Number 17Critical values when σ is unknownExampleExample (Continued)Slide Number 21Slide Number 22Health Behavior Statistical Methods HP 340L Lecture 8 Part II Using Statistics for Inference and Estimation Chapter 7 We covered material in Kiess & Green Chapter 7  as an estimator of the population mean µ  Estimation error  Mean and variability of  Distribution of as sample size increases Last Lecture xxx We would learn more about Confidence Interval Can think of estimating the mean as target shooting  Bull’s-eye is like the true mean µ  Shot is like our best guess of µ Estimating the Population Mean xWe want to ensure The estimator is • Unbiased, accurate, consistent and preciseHP-340: Fall 2016 5 Confidence Intervals• We usually want to know an estimate of the parameter. • We choose a single number (statistic) as our “best guess” of the unknown quantity (this is called a point estimate). – Usually use a sample statistic for the point estimate. – For example, is a point estimate μ Point Estimate x Interval estimation of the population mean  Confidence Interval (CI): Range of score values expected to contain the population mean µ with a certain ‘level of confidence’  Example: 46 ± 3 = (43,49) is a (95%) CI  Confidence Limits: The lower and upper scores defining the confidence interval  Example: Lower limit = 43; upper limit = 49 Confidence Intervals• Because there is variability (randomness) in how we select the members of the population to observe in our sample, there is variability/randomness/uncertainty in our estimate of the parameter. • We report a confidence interval which gives a range of plausible values of the population parameter. Interval Estimate• confidence intervals = – The standard error typically depends on: • The sample size n • The variability of the raw data – The constant is a theoretical value. It depends on: • The probability distribution of the test statistic • The chosen confidence level α. Confidence Interval (Constant)(Point est (Standaridmate e) rror)±×General Confidence Interval for a Mean • Formula: – Describes with 100(1 – α)% confidence where the true (population) mean μ lies. 1 /2( 1,1 /2) for kn foown r unknown nxznsxtnαασσσ−−−±⋅±⋅• The quantity z1–α/2 is called a critical value for the normal distribution. It represents the point for which there is an area of 1 – α/2 under the standard normal curve (μ = 0, σ = 1) to the left of z1–α/2 (and therefore an area of α/2 to the right). Critical Value• The quantity z1–α/2 is called a critical value for the normal distribution. It represents the point for which there is an area of 1 – α/2 under the standard normal curve (μ = 0, σ = 1) to the left of z1–α/2 (and therefore an area of α/2 to the right). Critical Value• The quantity z1–α/2 is called a critical value for the normal distribution. It represents the point for which there is an area of 1 – α/2 under the standard normal curve (μ = 0, σ = 1) to the left of z1–α/2 (and therefore an area of α/2 to the right). Critical ValueCritical values for confidence intervals for a mean (known σ) Interval α α/2 z1–α/2 80% Confidence Interval 0.20 0.10 1.282 90% Confidence Interval 0.10 0.05 1.645 95% Confidence Interval 0.05 0.025 1.960 99% Confidence Interval 0.01 0.005 2.576Example: 95% confidence interval for a mean (known σ) • IQ for sample of Los Angeles residents n = 7, = 99.6, σ = 15 • 95% Confidence Interval for Mean (known variance): – “We are 95% confident that the true population mean IQ of Los Angeles residents is between 88.5 and 110.7.” Sample mean known σ x151.96 99.6 11.1799.6 ± ⋅= ±Z0.975HP-340: Fall 2016 17 Std. normalt(1)t(5)t(10)Normal distribution -large n or -small n, known σ 2 t-distribution -small n -unknown σ 2 Distribution for a continuous variableCritical values when σ is unknown • When σ is unknown, we use the quantity t(n–1,1–α/2) for the critical value (instead of z1-α/2) • The quantity t(n–1,1–α/2) is a critical value for the t-distribution. It represents the point for which there is an area of 1 – α/2 under the graph of the t-distribution with n – 1 degrees of freedom to the left of t(n–1,1–α/2) (and therefore an area of α/2 to the right).Example Question: What is the mean pulse rate of students taking a midterm for HP 340? Design experiment: – 90 students in the class – Each TA selects 10 students at random and measures pulses For 1st TA, the pulse rates were – 71, 72, 74, 74, 75, 75, 76, 77, 78, 80 beats per minute For the 2nd TA, the pulse rates were – 70, 73, 74, 75, 76, 76, 77, 79, 80, 81 beats per minuteExample (Continued) • 95% confidence interval for mean (unknown σ) • n = 10, so degrees of freedom, df = 10 – 1 = 9 • Using Table A.4, the critical value t(9,0.975) = 2.262 • Confidence intervals are – For TA 1: – For TA 2: ( 1,1 /2)nsxtnα−−±⋅2.7075.2 2.262 75.2 1.9 (73.3,77.1)10± ⋅ = ±=3.3576.1 2.262 76.1 2.4 (73.7,78.5)10± ⋅ = ±= Width of CI tells us how accurately we know the population mean at a given level of confidence, e.g., at 95%; a narrower interval indicates better accuracy  As n increases:  SE of the mean decreases  Width of CI decreases  Confidence level stays at 95%  As confidence level increases (from 95% to 99%) width increases Confidence