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Discussion section 227
11/09/10
Confidence Intervals
Point estimates vs. interval estimates
•
Estimates of population parameters can be either point estimates or interval estimates.
•
A point estimate (e.g. sample mean) is precise but likely inaccurate
•
An interval estimate is less precise but likely more accurate. It is a range of sample statistics we
could expect to have if we repeatedly conducted hypothesis tests using samples from our
population.
Confidence interval
: an interval estimate based on our
sample mean
that includes the population mean
a certain percentage of the time (e.g. 95%) if we were to sample from the same population repeatedly.
Calculating confidence intervals:
Z distribution
Remember that we use a z test when we know that SD and M of the population.
Example: Let’s say μ = 72, σ = 10, M = 75, n = 40 and we want to calculate a confidence interval around
this mean.
1. Draw a sampling distribution and put the
sample mean
in the middle, not the population mean. Write
the percentage area under the curve.
2. In your z table, find the area under the curve corresponding to 47.5% and write the corresponding z
scores (±1.96 as you may remember).
3. Now, turn the z scores into raw scores.
•
It is important to remember that we are calculating an interval around our
sample mean
and so
we use it in our calculation – not the population mean.
•
Second, because we have a sample mean rather than an individual score, we use a distribution
of means and so we need to calculate the standard error: σ/√n = 10/√40 = 1.58
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Next, use the Z score formula to find the value for the sample means at each end of the
distribution. Remember that our original Z score formula was:
Now we are solving for M
(lower)
and M
(upper)
and using M
(sample)
instead of the population mean:
Our confidence interval is then reported in parenthesis (71.9, 78.1).
To summarize:
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 Fall '10
 Fairchild

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