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Unformatted text preview: Chapter 8 Statistical Inference and Sampling 2 Normal Curve for Population Individual observations, X’s, follow a normal distribution with mean = μ and standard deviation = σ. The following figure portrays the shape of normal population. σ x μ That is, X is a normal random variable. The corresponding standard normal variable Z can be obtained by the following. σ μ = X Z 3 Examples on Normal Curve for Population The estimated milespergallon ratings of a class of trucks are normally distributed with a mean of 12.8 and a standard deviation of 3.2. What is the probability that one of these trucks selected at random would get between 13 and 15 miles per gallon? σ μ = X Z 15 12.8 13 X z2 z1 z 06 . 0625 . 2 . 3 8 . 12 13 1 2245 = = z Or, the area from mean to z1 = 0.0239 0236 . ) 1 ( , = ≤ ≤ z z P So 69 . 6875 . 2 . 3 8 . 12 15 2 2245 = = z Or, the area from mean to z2 = 0.2549 2549 . ) 2 ( , = ≤ ≤ z z P So ? ) 15 13 ( = ≤ ≤ X P ? ) 2 1 ( = ≤ ≤ z z z P Or, the area from z1 to z2 = ? So, the area from z1 to z2 = 0.2549 – 0.0239 = 0.231 4 Examples on Normal Curve for Population The examination committee of the American Society for Quality passes 40% of those that take the exam. If the scores follow a normal distribution with an average score of 75 and a standard deviation of 16, what is a minimum passing score? σ μ = X Z 16 . 79 16 . 4 75 ) 26 . )( 16 ( 75 16 75 26 . = + = = = X X X 40 . % 40 ?) ( = = ≥ X P The area from mean to z = 0.50 – 0.40 = 0.10 So, z = 0.26 [From Normal Dist. Table] 75 X X 40% 40 . % 40 ?) ( = = ≥ z P z Z 40% 5 Estimation Statistical estimation is the process of estimating a parameter of a population from a corresponding sample statistic . Example: Usually population means (μ) are unknown and have to be estimated from sample means ( ). Two Approaches to Statistical Estimation Point estimate: A single value that represents the best estimate of the population value. For example, the sample mean ( ) is the best point estimate for the population mean (μ). Similarly, the sample standard deviation (s) is the best point estimate for the population standard deviation (σ). That is, μ = Xbar, and σ = s. Interval estimation: Builds on point estimate to arrive at a range of values that we are confident contain the population parameter. The range of values is called a confidence interval. For example, the confidence interval for population mean (μ LL ≤ μ ≤ μ UL ) can be estimated from the sample mean. X X Xbar μ UL μ LL Note that μ LL and μ UL are equidistant from Xbar, and are estimated from Xbar 6 Distribution of Xbar Xbar is a random variable, because different samples drawn from the same population on a specific characteristic will result in different values of Xbar....
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This note was uploaded on 04/15/2011 for the course DSCI 2710 taught by Professor Hossain during the Spring '08 term at North Texas.
 Spring '08
 Hossain

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