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Confidence intervals

- www.ibmaths.com

These intervals are for distributions where the

standard deviation of the population is known.

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,

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.

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.

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 .

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.

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