This is Stephanie from StatisticsHowTo.com and in this video I will show how to construct
a confidence interval with a t-distribution. I have a sample problem here.
We are being asked to construct a 98% confidence interval based on a set of data.
This is the formula we are going to use. It is x bar plus or minus the t score and
I will show you how to find that in a moment. Multiplied by s, that is the standard deviation
of a sample. Divided by the square root of n.
n is the number of items in our sample. First, we need to know the sample mean.
That is our X bar. If we add all of these up and then divide
by 10, we get a mean of 71. And the standard deviation for the data.
I am not going to go into how to find the standard deviation in this video, but if you
would like to click the link showing on screen, it will take you to an article explaining
how to find the standard deviation and also a standard deviation calculator.
Standard deviation for this particular sample is 18.172.
So we have found our s and we found our X bar.
Next we want to find this T score. We need a couple of steps to find the t score.
The first is we need to find the degrees of freedom.
I will just write t score here. So to find the degrees of freedom, we subtract
1 from our sample size. We have 10 numbers in our sample.
If I subtract 1, I get 9. The second step we need to find the t score
is we need to subtract the confidence level from 1.
It is 98%, so if I subtract .98 then I want to divide this number by 2.
That gives us the alpha level. I get .01.
I need these two numbers to look up the t score in the t-distribution table.
So I am looking for a degrees of freedom of 9, and an alpha for level of .01.
Here is my alpha level .01 and if I look down to 9 degrees of freedom.
Here is the degrees of freedom in this column. I get 2.821 , so my t score is 2.821.
Now, we have everything we need for the formula. We are actually going to work this part of
the formula here. This t times s divided by the square root
of n. So I have got t times s, the square root of
n that gives me 2.821. So my t score, multiplied by s, 18.172, divided
by the square root of 10, and I am going to figure this out on a calculator.
That comes out to 16.22075.
Now the confidence interval tells us to take
the mean and plus or minus this figure. So I am going to take the mean 71.
And I am going to subtract 16.22075. And separately I am also going to add 16.22075.
So this is our plus or minus.
And 71 minus 16.22075 is 54.77925.
And if I add this number to 71, I get 87.22075. This is the lower level for the confidence
interval and this is the upper level for the confidence interval.
That is how to find the confidence interval with a t-distribution.
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