Question

How do you find the 95% confidence interval?

Find the 95% confidence interval for a sample of size 39 with a mean of 20.3 and a standard deviation of 10.2.

Answer 1

A 95% confidence interval with the given data gives: `(17.1, 23.5)`

Explanation:

The confidence interval is given by the formula:

`barx +-z(sigma/sqrtn)`

Or you can write it as:

`(barx -z(sigma/sqrtn),barx +z(sigma/sqrtn))`

Where `barx` is the sample mean, `z` is the standardized value for the normal distribution, in relation to the percentage (don't really know how to explain this one that well), `sigma` is the standard deviation, and `n` is the sample size.

We know all the values except for `z`. To find the `z` value, we can imagine a normal distribution graph with 95% of it shaded, where the middle of this is the mean.

![https://en.wikipedia.org/wiki/1.96]()

As you can see from this picture the `z` value is 1.96. This can be found by using a normal distribution percentage points look up table.
I hope you can see that to the left and right of the shaded area, 2.5% is taken up by the white space each side.

So therefore you do 95%+2.5% = 97.5% then you can look that value up in the tables which is in fact: 1.96.

Now you can just substitute all the numbers into the expression:

`(20.3 -1.96(10.2/sqrt39),20.3 +1.96(10.2/sqrt39))`

Enter this into your calculator and you get:

`(17.1, 23.5)`

Hope this helps!