6.1
Estimating with Confidence
those who took the SAT math in 2003:
mean= 519, sd= 115
give test to SRS of 500 California students:
x
= 461 (math)
What can you say about the mean score μ
in population of 385,000?
•
x
is an unbiased estimator of μ (
x
μ
= μ)
• but how reliable is this estimate?
(how variable is the statistic?)
recall
x
is approximately N (μ,
n
σ
) when n is large
suppose σ = 100, then
x
=
4.5 (unrealistic to assume we would know σ CH 7)
x
has normal distribution centered at unknown population mean μ with
x
=
4.5
Figure 6.2
•the 689599.7 rule says that, for any N distribution, 95% of observations within 2 sd of mean
•95% chance→
x
will be within 2 standard deviations of
x
•95% chance→
x
will be within 9 points of population mean μ
•to say that
x
lies within 9 points of μ is the same as saying that μ is within 9 points of
x
•so 95% of all samples will capture the true μ in the interval from
x
 9
to
x
+ 9
our sample gave
x
= 461
95% confidence interval=
x
±
9
461
±
9
[452, 470]
we say that we are 95% confident that the unknown mean score lies between [452, 470]
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View Full Documentour sample gave
x
= 461
95% confidence interval=
x
±
9
461
±
9
[452, 470]
we say that we are 95% confident that the unknown mean score lies between [452, 470]
Once we look at a sample and construct a confidence interval, there are two possibilities:
1) The interval between 452 and 470 contains the true population mean μ
Our sample is one of the 95% of samples where μ lies within 9 points of
x
.
2) The true population mean μ is not contained between 452 and 470 (rare result).
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 Spring '08
 ABDUS,S.
 Normal Distribution, Standard Deviation, σ σ

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