Confidence intervals

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Confidence intervals

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These intervals are for distributions where the
standard deviation of the population is known.
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Confidence intervals for the population mean
From previous work on the Central Limit Theorem
we know that if a sample is taken from any type
of statistical distribution then the sample will
be distributed normally as,
From this we can calculate standardised Z values
and produce confidence intervals by using the
calculation,
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Producing a 95 confidence interval
A 95 confidence interval looks like this.
The above calculation gives us two limits within
which the population mean lies. Further the
probability that the mean lies within these
limits is 95 certain, a probability of 0.95.
Producing the general equation.
The limits can be summarised as follows.
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Example 1
The heights of children in a primary school are
known to have a variance of 64 cm. A sample of 36
children are taken the school and the mean
average height was calculated as 132 cm. The
population is known to be normal. Produce a 99
confidence interval for the population mean.
Find the z values.
-2.58
2.58
The z values can be found from the normal tables
or from a GDC.
The final confidence interval is (128.56,135.44)
From a GDC Menu,Stat,F5,F1,F3. 0.995,1,0
There is a 0.99 probability that the mean of the
population lies between 128.56 and 135.44.
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Sampling a binomial distribution and finding the
confidence interval of proportion.
From previous work we have seen that we can
approximate a binomial distribution to a normal
distribution if n is large enough.
If XB(n,p) and n is large then XN(np,npq).
When we evaluate confidence intervals for
binomial distributions we often give the
proportion of the population. This must be a
value between 0 and 1.
So the sample mean becomes , this is
sometimes called .
So the sample variance is .
And the standard error that is used in the
calculation is .
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Example 2 a binomial distribution
A survey is conducted to test whether students at
a school like the new maths teacher. 100 students
are surveyed and 30 say they like the new teacher
and 70 do not. Produce a 95 confidence interval
for the proportion of the population that like
the new teacher.
From the sample we need to get some statistics.
This is a binomial of we have
The estimated mean is given by p.
The standard error is given by
The standard error
0.2102 and 0.3898
Confidence interval for the proportion of the
population that like the new maths teacher is
(0.2102, 0.3898)
Now draw the normal curve and find the z values.