Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
Chapter 4 Some Elementary Statistical Inferences 4.1 SamplingandStatistics In Chapter 2, we introduced the concepts of samples a ...
226 Some Elementary Statistical Inferences In this process, our information about the unknown distribution ofXor the unknown par ...
4.1. Sampling and Statistics 227 This is called thelikelihood functionof the random sample. As an estimate of θ, a measure of th ...
228 Some Elementary Statistical Inferences realized values are: 359 413 25 130 90 50 50 487 102 194 55 74 97 For instance, 359 h ...
4.1. Sampling and Statistics 229 The two partial derivatives simplify to ∂l(μ, σ) ∂μ = − ∑n i=1 ( xi−μ σ )( − 1 σ ) (4.1.5) ∂l(μ ...
230 Some Elementary Statistical Inferences Example 4.1.4(Uniform Distribution). LetX 1 ,...,Xnbe iid with the uniform (0,θ) dens ...
4.1. Sampling and Statistics 231 Next, suppose that the space ofXis infinite, say,D={a 1 ,a 2 ,...}. In practice, we select a va ...
232 Some Elementary Statistical Inferences Medium Fair Dark Red Black Haircolor 0.0 0.1 0.2 0.3 0.4 Bar Chart of Haircolor of Sc ...
4.1. Sampling and Statistics 233 Histogram of Poisson Variates Number of events 0123456 024681 0 Figure 4.1.2:Histogram of the P ...
234 Some Elementary Statistical Inferences The histogram provides a somewhat crude but often used estimator of the pdf, so a few ...
4.1. Sampling and Statistics 235 lines(density(sulfurdioxide)) y=dnorm(sulfurdioxide,53.91667,10.07371);lines(y~sulfurdioxide,lt ...
236 Some Elementary Statistical Inferences the histogram. Use the R functiondgamma(x,shape=1,scale=θˆ)to evaluate the pdf. (c)Ob ...
4.1. Sampling and Statistics 237 (b)Suppose the sampling continues untilX 1 is no longer the smallest observation (i.e.,Xj<X ...
238 Some Elementary Statistical Inferences (d)Obtain the sample mean and standard deviation and on the histogram overlay the nor ...
4.2. Confidence Intervals 239 That is, the probability that the interval includesθis 1 −α, which is called the confidence coeffi ...
240 Some Elementary Statistical Inferences In Example 4.1.3, we presented a data set on sulfur dioxide concentrations in a damag ...
4.2. Confidence Intervals 241 In practice, we often do not know if the population is normal. Which confidence interval should we ...
242 Some Elementary Statistical Inferences LetS 12 =(n 1 −1)−^1 ∑n 1 i=1(Xi−X) (^2) andS 2 2 =(n 2 −1) − 1 ∑n 2 i=1(Yi−Y) (^2) b ...
4.2. Confidence Intervals 243 result of Section 3.6.1 concerning Student’st-distribution, we have that T = [(X−Y)−(μ 1 −μ 2 )]/σ ...
244 Some Elementary Statistical Inferences n=n 1 +n 2 be the total sample size. Our estimator ofp 1 −p 2 is the difference in sa ...
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