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Everything that we did in the last four chapters is related to this chapter. However, the logic of what we are doing here, estimation, is different from the logic used in hypothesis testing.
In the last several chapters we tested the a null hypothesis that basically asked the question, is this different from that? Estimation asks a different question. With estimation we are making educated guesses as to the value of a population parameter.
When do we use estimates?
We'll focus on two kinds of estimates of the population mean.
2) interval estimates (confidence intervals) of the mean: using a range of values as your estimate of an unknown quantity. When an interval is accompanied with a specific level of confidence (or probability) , it is called a confidence interval.
Both kinds of estimates are determined by the same equation, the difference is that for the point estimates, we'll just compute a single number (that's why it is called a point estimate), but for the interval estimate, we'll compute an interval between two points.
Let's start at the conceptual level. Consider the following population distribution.
Suppose that we guess that the mean is somewhere between 71 & 99? How confident are we in this guess?
point estimate | interval estimate | |
Disadvantages | it doesn't convey any sense of how much precision we have in making that estimate. | we often need to have one specific value, a range of possible values just may not be enough |
Okay, now let's begin with a point estimate of the mean. What will be the best single estimate of the population mean?
population sample means
However, suppose that all we have is a single sample. Now what is our best guess?
How can we get an estimate where we'd have a better chance of being right? Instead of giving a point estimate, we can estimate an interval.
Okay, now let's formalize things a bit. Let's first talk about the logic of estimation, and then move onto the actual formulas that we'll use.
Okay, so what's the formula? It is the same one(s) that we've been using all along, but we do a little algebra to move it around so that instead of solving for a z-score, we solve for the population parameter.
z = ---> (z)() = - m ---> m = - ()(z)
step 2: and we see that m = is our most reasonable estimate.
Okay, so that's the formula for point estimation. What about for an interval estimation?
m = + (z)()
= 85 + (1.65)(5/sqroot 25) = 86.65
m = - (z)()
= 85 - (1.65)(5/sqroot 25) = 83.35
The above example, used z-scores. The same logic applies to using t-statistics. How do you know which test statistic to use? Same logic as prior chapters. It depends on the design.
The other formulas for estimation:
independent samples: m = (1 - 2) + (tcrit)(sx1-x2)
related samples: m = D-bar + (tcrit) (est std error of D)
Note: when using t-tests, make sure that you use the appropriate dfs