Inference on Single Mean

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Unformatted text preview: na; Slide 29 University 29 Can We Assume Sampling from a Normal Population? If data are from a normal population, there is a linear relationship between the data and their corresponding Z values. data Z= Y −µ σ Y =σ Z +µ If we plot y on the vertical axis and z on the horizontal axis, the y intercept estimates µ and the slope estimates σ. G. Baker, Department of Statistics G. University of South Carolina; Slide 30 University 30 How to Calculate Corresponding How Z-Values Order data Order Estimate percent of population below each data point. data i − 0.5 Pi = n where i is a data point’s position in the ordered set and n is the number of data points in the set. Look up Z-Value that has Pi proportion of Look proportion distribution below it. distribution G. Baker, Department of Statistics G. University of South Carolina; Slide 31 University 31 Normal Probability (QQ) Plot Data set: Z Pi yi i 7 1 -0.32 10 2 .375 4 2 .625 7 3 +1.15 4 .125 +0.32 2 -1.15 .875 10 4 Normal QQ Plot 12 10 Data 8 6 4 2 0 -1.5 -1 -0.5 0 0.5 1 1.5 Z values G. Baker, Department of Statistics G. University of South Carolina; Slide 32 University 32 Normal Probability (QQ) Plot QQ Plot with Data on Vertical Axis 16 14 12 10 8 6 4 2 0 -3 -2 -1 0 1 2 This data is a random sample from a n(10,2) population. G. Baker, Department of Statistics G. University of South Carolina; Slide 33 University 33 3 Normal Probability (QQ) Plot QQ Plot with Data on Vertical Axis 16 14 12 10 8 6 4 2 0 -3 -2 -1 0 1 2 3 G. Baker, Department of Statistics G. University of South Carolina; Slide 34 University 34 Estimation of the Mean G. Baker, Department of Statistics G. University of South Carolina Point Estimators A point estimator iis a single number s point calculated from sample data that is used to estimate the value of a parameter. estimate Recall that statistics change value upon repeated sampling of the same population while parameters are fixed, but unknown. parameters Examples: Examples: ˆ p estimates p ˆ µ = y estimates µ ˆ ˆ = s estimates σ σ 2 = s 2 estimates σ 2 σ G. Baker, Department of Statistics G. University of South Carolina; Slide 36 University 36 In General: θˆ is an estimator of the arbitrary parameter θ What makes a “Good” estimator? (1) Accuracy: An unbiased estimator of a parameter is one whose expected value is equal to the parameter of interest. (2) Precision: An estimator is more precise if its sampling distribution has a smaller standard error*. *Standard error is the standard deviation G. Baker, Department of Statistics G. for the sampling distribution. University of South Carolina; Slide 37 University 37 Unbiased Estimators For normal populations, both the sample mean and sample median are unbiased estimators of µ. Sampling Distributions for Mean and Median mean median -8 -6 -4 -2 0 µ 2 4 6 8 G. Baker, Department of Statistics G. University of South Carolina; Slide 38 University 38 Most Efficient Estimators If you have multiple unbiased estimators, then you choose the estimator whose sampling distribution has the least variation. This is called the most efficient estimator. effi...
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