Point estimates often need to be nested in layers of analysis, and it is the Inv
ID: 3368218 • Letter: P
Question
Point estimates often need to be nested in layers of analysis, and it is the Invariance Principle that provides a pathway for doing so. An example would be estimating Mu after having to estimate Alpha and Beta for some distributions. In simpler statistics class exercises (like those we've seen up until now), this is typically avoided by providing the lower level parameters within exercises or problems (e.g. asking you for Mu by giving you the Alpha and Beta). The only real options we've had prior to this unit for estimating lower level parameters has been trial-and-error: collecting enough data to form a curve that we then use probability plots against chosen parameter values until we find a combination of parameters that "fits" the data we've collected. That approach works in the simplest cases, but fails as our problem grows larger and more complex. Even for a single distribution (e.g., Weibull) there are an infinite number of possible Alpha-Beta combination. We can't manually test them all. Point estimation gets us around all of that by providing the rules needed to actually calculate lower level parameters from data. We sometimes need to be able to collect a lot more data to use this approach, but it's worth it. We'll be able to calculate more than one possible value for many parameters, so it's important that we have rules for selecting from among a list of candidates. Discuss what some of those rules are, and how they get applied in your analysis. If an engineering challenge includes "more than one reasonable estimator," (Devore, p. 249) how do engineers know which to pick, and what issues arise statistically and in engineering management when making those choices?Explanation / Answer
(a) THE USE OF THE STANDARD DEVIATION FOR PREDICTION
it stated that specific predictions could be made about how many numbers in a set would fall within a standard deviation or standard deviations from the mean.
The empirical rule predicts the following:
1. Approximately 68% of all numbers in a set will fall within ±1 standard deviation of the mean.
2. Approximately 95% of all numbers in a set will fall within ±2 standard deviations of the mean.
3. Approximately 99% of all numbers in a set will fall within ±3 standard deviations of the mean.
Application-(i) AS A DEFINITION OF AN OUTLIER-e, a few statisticians define an outlier as a score in the set that falls outside of ±3 standard deviations of the mean.
(ii)PREDICTION AND IQ TESTS-IQ scores (on Wechsler’s IQ tests) have a theoretical mean of 100 and a standard deviation of 15. Therefore, we can predict with a reasonable degree of accuracy that 68% of a random sample of normal people taking the test should have IQs between 85 and 115.
(b)
Statisticians use sample statistics to estimate population parameters. For example, sample means are used to estimate population means; sample proportions, to estimate population proportions.
An estimate of a population parameter may be expressed in two ways:
(c) properties to make a good point estimator
1. It’s desirable that the sampling distribution be centered around the true population parameter. An estimator with this property is called unbiased.
2. It’s desirable that our chosen estimator have a small standard error in comparison with other estimators we might have chosen.
Sample Proportion ± 1.96 × Standard Error
(d)
Statisticians use a confidence interval to express the precision and uncertainty associated with a particular sampling method. A confidence interval consists of three parts.
The confidence level describes the uncertainty of a sampling method. The statistic and the margin of error define an interval estimate that describes the precision of the method. The interval estimate of a confidence interval is defined by the sample statistic + margin of error.
For example, suppose we compute an interval estimate of a population parameter. We might describe this interval estimate as a 95% confidence interval. This means that if we used the same sampling method to select different samples and compute different interval estimates, the true population parameter would fall within a range defined by the sample statistic + margin of error 95% of the time.
Confidence intervals are preferred to point estimates, because confidence intervals indicate (a) the precision of the estimate and (b) the uncertainty of the estimate.