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.