Let’s look at finding the appropriate z value

using the standard normal table

when constructing a confidence interval.

Suppose we wish to use this confidence interval formula

to obtain a (1-alpha) times 100% confidence interval for mu.

This would be the appropriate formula if we were sampling

from a normally distributed population

where the population standard deviation sigma is known.

And the subject of this video is how to

find this z sub alpha/2 value

using a standard normal table.

The z value is based on the standard normal distribution,

and I’ve plotted in the standard normal distribution here.

And if we want the confidence level to be (1-alpha) times 100%,

we put an area of 1-alpha in the middle of the distribution.

The part that’s left over is alpha,

because the area under the entire distribution is 1.

And we split this alpha evenly into the two tails,

putting alpha over 2 in the right tail

and alpha over 2 in the left tail.

z_alpha/2 is the z value

that has an area to the right I of alpha/2.

Let’s look at a specific example.

What is the appropriate z value for a 95% confidence interval?

Well, here we’re saying that

(1-alpha) times 100% is 95%

and we put 95% of the area, or .95,

in the middle here.

So this area is going to be 0.95.

alpha is the remaining area, 0.05

and we split that evenly into the two tails,

putting an area of 0.025 in the right tail

and an area of 0.025 in the left tail.

Now we need to find this value

that has an area to the right of 0.025 under the standard normal curve.

And we have to remember that

the standard normal table that I use

gives the area to the left, so the table gives the area to the left,

and so when we go to the table,

we’re going to have to recognize that

we need to look up the area to the left.

and the area to the left of the value we need is 0.975.

So let’s go to the table and find that value.

Here’s my standard normal table, which gives areas to the left.

and we need this area to be 0.975.

Areas are found in the body of the table here,

so we’re going to have to look in here

and get as close as we can to 0.975.

And right here we find 0.9750,

and so we go to the edges of the table to find 1.96,

and so the appropriate z value is 1.96.

And so the table tells me

that this value right here is 1.96

or in other words, z_0.025,

the z value with an area of 0.025 to the right, is 1.96.

This value over here, due to symmetry about 0 is -1.96,

and so our confidence interval

will be found by taking the value of the sample mean

and adding and subtracting

1.96 times sigma over the square root of n.

What is the appropriate z value for a 75% confidence interval?

Well here we’re going to put a middle area of 0.75,

and alpha, the remaining bit is 0.25.

And alpha/2 is 0.125

and so we’re going to put this area of alpha/2

in each of the tails, 0.125 and 0.125.

And now we need to find this value.

Well this value has an area to the right of 0.125

and an area to the left of 0.875

So let’s go to the table.

We want to find the z value

that has an area to the left of 0.875,

and so we look in the body of the table

and get as close as we can to 0.875,

and that’s 0.8749 right here, and so

we go to the edges to get our z value, 1.15.

And so the z value that we need is approximately 1.15.

So the table tells me that this value

to two decimal places is 1.15

and this value, by symmetry about zero is -1.15

and so our z_.125 is approximately 1.15,

and the appropriate 75% confidence interval

is going to be X bar plus and minus

1.15 times sigma over the square root of n.