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Point estimation, a statistical method used to estimate unknown population parameters using sample statistics. the concepts of point estimates and unbiased estimators for population mean and proportion, using examples for IQ levels and opinions on legalizing marijuana. Probability theory is used to explain why sample mean and sample proportion are good choices as point estimators. The document also mentions the importance of random sampling and unbiased study designs.
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{Estimation} We an unknown begin with quantity. a discussion of point estimation where we use a single number to estimate
We already know how to find point estimates but in this section we will formalize a few properties of good point estimates and discuss their limitations.
Suppose we are interested in studying the IQ levels of students at Smart University (SU). In particular mean IQ level (since of IQall levelthe students is a quantitative at SU. A variable),random sample we are of interested 100 SU students in estimating is taken μ , theand the (sample) mean IQ level (x‐bar) was found to be 115. If we wanted to estimate μ , the population mean IQ level, by a single number based on the sample, sample meanit would which make is 115 intuitive sense to use the corresponding quantity in the sample, the
We mean say (x that‐bar) (^115) as theis the point point estimator estimate for for μ μ , and in general, we'll always use the sample
Note: talk in when general we about talk about the statistic the specific , the (^) random value (115), variable we use x‐bar, the we term use estimate the term, and estimator. when we
Also, here we are still in a less realistic scenario as we are assuming we know sigma, the population will not have standard this luxury. deviation. We are Although approaching for SAT these scores problems this may this be way realistic, initially in so general that we we can use the normal distribution in our demonstration of these concepts. Later, we will look at approximate how to handle it from the our more data. realistic situation when sigma is not known and we must
Point estimation is very intuitive, certainly, our intuition tells us that the best estimator for μ should be x‐bar, and the best estimator for p should be p ‐hat Probability theory does more than this; it actually gives an explanation (beyond intuition) respectively why x‐bar and p ‐hat are the good choices as point estimators for μ and p ,
From random our, thestudy distribution of sampling of distributionssample means we is foundexactly that centered as long at as the a sample value of is population taken at mean X‐bar is therefore said to be an unbiased estimator for μ Any particular sample mean might turn out to be less than the actual population mean, or it they might will turn not outunderestimate to be more, any but more in the or long less run, often such than sample they overestimatemeans are "on target" in that
We unbiased have discussed estimator biasis that a few the times mean from, also a called logical perspective the expected (^) value but the, of true the definition statistic (the of an mean of the sampling distribution) is equal to the target population parameter.
Likewise, we learned that the sampling distribution of the sample proportion, p ‐hat, is centered making p (^) ‐athat the an population unbiased estimator proportion for p (^) p (as. long as the sample is taken at random), thus
Probability We stated that theory probability plays an wasessential the foundation role as we and establish sampling results distributions, for statistical the inference. bridge, to statistical inference. Our assertion above that sample mean and sample proportion are unbiased estimators is our first step on that bridge. The theory. definition of an unbiased estimator is a statistics definition that relies on probability
There the population are many has other a sample examples counterpart of this idea. that Any we parametercan study in I maya similar want way. to estimate from
Not only are sample mean and sample proportion on target as long as the samples are random, precision but improves they become as sample less size variable increases. as the sample size increases, in other words, their
We have equations for the standard error in the two cases we are currently considering.
Another example of a point estimate is using sample standard deviation (s) to estimate population standard deviation, σ We will not be concerned with estimating the population standard deviation for its own sake, standardizing but since the we sample will often mean, substitute it is worth the pointingsample standard out that deviation s is an unbiased ( s ) for σestimator when for σ an, in unbiased fact the estimator!reason that we divide by n‐ 1 instead of n is because dividing by n‐ 1 results in
When we estimate μ by the sample mean x‐bar we are almost guaranteed to make some kind of error! Even though we know that the values of x‐bar fall around μ , it is very unlikely that the value of x‐bar will fall exactly at μ. Given themselves that such of limited errors usefulness,are a fact of unless life for we point are estimates,able to quantify these theestimates extent areof the in estimation error. My favorite analogy for estimation is the reverse game of darts! The on the parameter wall. we are trying to “hit” with our estimate is a single number, the tip of a dart
Now consider trying to hit the tip of a dart with another dart! That is point estimation! It is very unlikely, no matter how good we are at throwing darts, that we can hit it exactly.
Interval estimation addresses this issue. The supplying idea behind information interval about estimation the size is, of therefore, the error (^) attached. to enhance the simple point estimates by
We want to quantify the potential error in using our estimate from our sample to represent the population value. From what we know about the sampling distributions of x‐bar and p‐hat combined with our ability population to work mean with μ normal) and the distributions, population proportionwe can construct ( p ) confidence intervals for the
Returning dart (my point to our estimate) analogy of at thethe fixedreverse tip gameof a dart of darts, on the now wall instead (my parameter), of throwing I get a single to throw the able whole to hit dartthe target. board (my interval estimate). Now we will increase our chances of being
In of (^) many throwing ways, a dart we say board statistics at a dart is backwards. is indeed very This backwards! analogy illustrates this well as the idea
Estimation is an important aspect of statistical inference. Even when we are conducting hypothesis tests, there will still be the need for estimation. We have discussed point estimates and the desired properties of being unbiased and less variable. We estimation. have discussed the drawbacks of point estimates and introduced the idea of interval
In intervals. the next section we will outline the process of creating and interpreting confidence
We use willdata see to estimatein the later an modules unknown that population confidence parameter, intervals even are useful when wheneverthis parameter we wish is to estimated using multiple variables (such as our cases: CC, CQ, QQ) For or the example, correlation we can coefficient. construct confidence intervals for the slope of a regression equation
In population doing so weparameter are always (the using TRUE our slope, data or to the provide TRUE ancorrelation interval estimate coefficient). for an unknown
