patients, a mean weight of 240 pounds is given, sample standard deviation is 25 pounds.
We want to find the confidence interval for a sample for a true mean weight of all foot
surgery patients. Just going to draw a quick sketch of what
we are looking for. Here's our mean, the 95% confidence interval
is going to be the bulk of everything in the middle here, and it’s going to be this area.
This little tail on this side is what we call alpha.
And this tail on the other side is also alpha. We find alpha by subtracting our confidence
interval from 100%, so 100% - 95% = 5%, this is our alpha level and we normally convert
this to a decimal. So 0.05, there is alpha in both tails, so we need
to divide this by two because I want to find just this one area here and you'll see
while in a moment. So if I divide 0.05 by two I get 0.025, I'm
going to look this up in the T-Distribution Table because I want to know what this value
is right here. We're using the T-Distribution because we
have a small sample of 10 patients. Anything below 30 for your sample size you
are going to want to use the T-Distribution. We need one more piece of information to look
up this value and that is the Degrees of Freedom. Well the Degrees of Freedom is just my sample
size which is 10 minus one. So I have 9 degrees of freedom and an alpha
of 0.025 and I'm going to pull up a T-Distribution Table.
So here's my T-Distribution Table, I'm looking at 9 degrees of freedom and alpha levels on
the top row 0.025, so a value of 2.262. So we found our T-value it's 2.262, I'm going
to call this step one, find the T-value. Step two, you want to calculate the standard
error, use the standard error on samples. It is a pretty simple formula; it is standard
deviation over the square root of n. Standard deviation was given up here it's 25
pounds, n our sample size was 10, 25 over the square root of 10.
If I work this on a calculator I get 7.90569415.
The next step, we want to multiply our T-value,
that was step one by our standard error in step two.
So I'm going to multiply 2.262 by 7.90569415, that was this figure right here.
When I do that, I get 17. 8826802.
Now what I have done is I have used the T-value,
this value right here to tell us where the mean lines we are looking for an upper value
for the mean up here and a lower value for the mean down there.
So we are going to add or subtract this to the mean to find the actual confidence
interval. So our final step, is to take the mean, it
is given in the question it's 240, I'm going to subtract this number 17.8826802, and
when I do that I get 222.117, I'll round it to three decimal places here in the final
step and also I want to add 17.8826802, this gives me the lower limit for the confidence
interval and this gives me the upper limit. That's how to find the confidence interval
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