Math 3339 Section 19377 MWF 10-11am SR 116 (and section 20582 – online) Bekki George [email protected] 639 PGH Office Hours: 11:00 - 11:45am MWF or by appointment Section 6.1 - Some General Concepts of Point Estimation Definition A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ. Example: Consider the accompanying 20 observations on dielectric breakdown voltage for pieces of epoxy resin. 24.46 25.61 26.25 26.42 26.66 27.15 27.31 27.54 27.74 27.94 27.98 28.04 28.28 28.49 28.50 28.87 29.11 29.13 29.50 30.88 The pattern in the normal probability plot given there is quite straight, so we now assume that the distribution of breakdown voltage is normal with mean value µ. Because normal distributions are symmetric, µ is also the median lifetime of the distribution. The given observations are then assumed to be the result of a random sample X1, X2, . . . , X20 from this normal distribution. Consider the following estimators and resulting estimates for µ : a. Estimator = X, estimate = x = Σxi /n = 555.86/20 = 27.793 b. Estimator = x! , estimate = x! = (27.94 + 27.98)/2 = 27.960 c. Estimator = [min(Xi) + max(Xi)]/2 = the average of the two extreme lifetimes, estimate = [min(xi) + max(xi)]/2 = (24.46 + 30.88)/2 = 27.670 d. Estimator = Xtr(10), the 10% trimmed mean (discard the smallest and largest 10% of the sample and then average), estimator = xtr(10)= 27.838 Each one of the estimators (a)–(d) uses a different measure of the center of the sample to estimate µ. Which of the estimates is closest to the true value? We cannot answer this without knowing the true value. A question that can be answered is, “Which estimator, when used on other samples of Xi’s, will tend to produce estimates closest to the true value?” We will shortly consider this type of question. Definition A point estimator is said to be an unbiased estimator of θ if E( θˆ ) = θ for every possible value of θ. If θˆ is not unbiased, the difference E( θˆ ) – θ is called the bias of θˆ . That is, θˆ is unbiased if its probability (i.e., sampling) distribution is always “centered” at the true value of the parameter. Proposition When X is a binomial rv with parameters n and p, the sample proportion pˆ = X / n is an unbiased estimator of p. No matter what the true value of p is, the distribution of the estimator will be centered at the true value. pˆ Proposition Let X1, X2, . . . , Xn be a random sample from a distribution with mean µ and variance σ 2. Then the estimator is unbiased for estimating σ 2. Proposition If X1, X2, . . . , Xn is a random sample from a distribution with mean µ , then X is an unbiased estimator of µ. If in addition the distribution is continuous and symmetric, then X and any trimmed mean are also unbiased estimators of µ. Definition The standard error of an estimator θˆ is its standard deviation σ θˆ = V (θˆ ) . It is the magnitude of a typical or representative deviation between an estimate and the value of θ . EMCF 17 1.The sampling distribution of a statistic is a. The probability that we obtain the statistic in repeated random samples b.The mechanism that determines whether randomization was effective c. The distribution of values taken by a statistic in all possible samples of the same sample size from the same population d.The extent to which the sample results differ systematically from the truth 2. A statistic is said to be unbiased if a. The survey used to obtain the statistic was designed so as to avoid even the hint of racial or sexual prejudice b.The mean of its sampling distribution is equal to the true value of the parameter being estimated c. Both the person who calculated the statistic and the subjects whose responses make up the statistic were truthful d.It is used for honest purposes only 3.The number of graduates at Johns Hopkins University is approximately 2000, while the number at Ohio State University is approximately 40,000. At both schools a simple random sample of about 3% of the undergraduates is taken. Which of the following is the best conclusion? a. The sample from Johns Hopkins has less sampling variability than that from Ohio State b.The sample from Johns Hopkins has more sampling variability than that from Ohio State c. The sample from Johns Hopkins has almost the same sampling variability than that from Ohio State d.It is impossible to make any statement about the sampling variability of the two samples since the students surveyed were different. 4.The sample statistic x is the point of estimate of a. the population standard deviation σ . b.the population median. c. the population mean µ d.the population mode. 5. Choose A