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Lecture

# Ch8.pdf

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Department
Statistics
Course Code
STAT151
Professor
Paul Cartledge

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Ch. 8 - Statistical Inference Def’n: Estimation is the assignment of value(s) to a population parameter based on a value of the corresponding sample statistic. An estimator is a rule used to calculate an estimate. An estimate is a specific value of an estimator. Note: in this chapter, always assuming an SRS. - Notation: Let - θ be a generic parameter. ˆ Let - θ be an estimator – a statistic calculated from a random sample Cons-equently, θˆ is an r.v. with mean E( θˆ) = µ θˆand std. dev. σ θ 8.1 Point and Interval Estimates Def’n: A point estimate is a single number that is our “best guess” for the parameter. Æ like a statistic, but more precise towards parameter estimation. An interval estimate is an interval of numbers within which the parameter value is believed to fall. Generic large sample confidence intervals: Def’n: A confidence interval (CI) for a parameter θ is an interval estimate of plausible values for θ. With a chosen degree of confidence, the CI’s construction is such that the value of θ is captured between the statistics L and U, the lower and upper endpoints of the interval, respectively. The confidence level of a CI estimate is the success rate of the method used to construct the interval (as opposed to confidence in any particular interval). The generic notation is 100(1 – α)%. Typical values are 90%, 95%, and 99%. Ex8.1) Using 95% and the upcoming method to construct a CI, the method is “successful” 95% of the time. That is, if this method was used to generate an interval estimate over and over again with different samples, in the long run, 95% of the resulting intervals would capture the true value of θ. Many large-sample CIs have the form: point estimate ± (critical value) × (standard error) where “point estimate” is a statistic θˆused to estimate parameter θ, ˆ “standard error” is a statistic σθˆused to estimate std. dev. of estimator θ , “critical value” is a fixed number z defined so that if Z has std. norm. dist’n, then P(-z ≤ Z ≤ z) = 1 – α = confidence level The product of the “standard error” and “critical value” is the margin of error. Note: critical value z often denoted by z ,α/2ere the notation reflects P(Z > z) = α/2. Ex8.2) if the confidence level is 95%, then α/2 = 0.025 and z = 1.96. 0.025 (diagram drawn in class) Table 8X0 – Critical values for usual confidence levels 100(1 – α)% α α/2 z α/2 90% 0.10 0.050 1.645 95% 0.05 0.025 1.96 99% 0.01 0.005 2.58 The estimator θˆ and its standard error σθ are defined so that, when the sample size n is sufficiently large, the sampling distribution of θ −θ σˆ θˆ is well approximated by the standard normal distribution. Consequently, ⎜ θ −θ ⎟ P⎜− z σˆ ≤ z⎟≈ −α ⎝ θˆ ⎠ Algebraic manipulation yields Pθ − zσ ≤θ ≤θ + zσˆ ˆ ) ≈1−α θˆ θ 8.2 Large Sample CI for Population Proportion Recall the 3 rules regarding the general properties of the sampling distribution of . Then, when n is large, a (1 – α)100% CI for p is p(1 − p) p ± zα /2 n Note that n being large also allows for the standard error to use since p is unknown. The interval can be used as long as (1) np ≥ 15 and n(1− p) ≥ 15, (2) the sample can be regarded as a random sample from the population of interest. Ex8.3) A survey of 1356 random adults asked them to pick out the funniest city name in a list. 923 chose “Keokuk”, 74 chose “Walla Walla”, and 359 chose “Seattle”. Let p be the proportion of all adults who would have answered “Seattle” had they been polled. Construct a 95% confidence interval for p. : e z i s e l p - Parama eter = p - n = 1356 - Estimate: p = 359/1356 ≈ 0.265 ˆ ˆ - Sta
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