Week5ConfidenceIntervals.pdf
Next, we are going to construct a 1-sample T confidence interval using a Mean
and SD.
1 – sample Confidence Interval Estimate for the Mean and an unknown σ. The
equation will look like this: �̅� ± 𝑇∗ ( 𝑆𝐷
√𝑛 )
Notice that we are using a T – critical value for this confidence interval because
we are referring to a sample and not an entire population.
Everything after the ± sign is called the Margin of Error. (Hopefully this sounds
familiar because we introduced it during Week 2)
Margin of Error (ME) = 𝑇∗ ( 𝑆𝐷
√𝑛 )
The Standard Error (SE) = ( 𝑆𝐷
√𝑛 ) (This is also a value you have seen during Week 2).
BUT, if you don’t have the Marin of Error you will need to calculate the T- critical
value. To calculate the T-critical value we will use the =T.INV( ) function in Excel.
For a 95% confidence interval, you will take 1 - .95 = .05. α = .05. But since this is
a confidence interval and we will need to add AND subtract from the mean we
will take .05/2 = .025. The new α = .025. But remember how Excel puts functions
in a less than form. To find the value we will use in the Excel function we will take
1 - .025 = .975. This is the alpha value you use in the Excel function.
Next, we need to find the degrees of freedom. The degrees of freedom (DF) = n –
1. Because the sample size is 10. DF = 10 – 1 = 9. Now that we have these two
values you can use the Excel function to find the T critical value.
In Excel hit the = sign, then T.INV( .975, 9), then close the parentheses, and hit
Enter.
The T-critical value is 2.262157
Now that we have all the values we need we can calculate the confidence
interval. But first let’s review the descriptive statistics we found during Week 2.
Highlighted in Yellow are the SE and the ME that were calculated using the
descriptive statistics tool using the Data Analysis ToolPak. We also see that the
ME is for a 95% confidence, BUT this can be changed and customized to the value
you want.
When you use the Data Analysis ToolPak and click on Descriptive Statistics when
the new window pops up and you check the box that says “Confidence Level for
Mean” the default is 95%, but if you wanted to change that to 90% you can.
Once you change this to the value you want click OK.
The new Margin of Error is highlighted in Yellow. This is a nice short cut you can
use to cut down on the amount of algebra you will be doing.
Going back to our 95% confidence interval using this equation.
�̅� ± 𝑇∗ ( 𝑆𝐷
√𝑛 )
The mean is still $25,650 and now we know the Margin of Error is $2,495.50331.
Plugging these into the equation we get
25,650 ± 2.262157∗ ( 3488.47308
√10 )
25,650 ± 2495.50331 25,650 – 2495.50331 = $23,155 -> rounded to the nearest dollar 25,650 + 2495.50331 = $28,146 -> rounded to the nearest dollar The 95% confidence interval is ($23,155, $28,146).
We are 95% confidence that the sample price of the cars will be between $23,155
and $28,146.
