Estimating an Unknown Mean: Building and Understanding a Confidence Interval
You will act as the analyst. You do not know the population. You get one sample, and from it you build a confidence interval for the population mean. Then a simulation shows what "95% confident" really means. Work through the steps in order. Each step unlocks the next one.
Business context (Doane Example 8.3). The MBA admissions office at a medium-sized public university wants to know the
average GMAT score of its whole applicant pool. Checking every applicant's file is costly, so an analyst draws a
random sample of applicants and uses it to estimate the true mean score μ.
The true value of μ stays hidden until Step 6.
1
What we know and what we don't
In real business work, the population is unknown. We cannot see every applicant's score, so we cannot compute μ directly. All we will have is one sample.
Population mean
μ = ?
parameter: what we want to estimate
Population std. dev.
σ = ?
parameter: usually unknown too
Population shape
?
we never see the whole pool
Parameters vs. statistics
Symbol
Meaning
Type
Do we know it?
μ
Population mean GMAT score
Parameter (fixed number)
Unknown: the target
σ
Population standard deviation
Parameter (fixed number)
Usually unknown
n
Sample size (number of applicants sampled)
Design choice
You choose it
x
Sample mean
Statistic (changes from sample to sample)
Computed from the sample
s
Sample standard deviation
Statistic (changes from sample to sample)
Computed from the sample
The plan: use the statistic x as our best single guess (the point estimate) for the parameter μ. Then add a margin of error so we report a range instead of one number.
2
Draw your one sample
Finish Step 1 to unlock.
Choose the sample size. Doane Example 8.3 uses n = 20, so start there.
Here are the GMAT scores of your 20 randomly selected applicants. This is all the data you have.
In Excel: paste into column A starting in cell A1 (one score per cell). Then compute:
the sample mean x with =AVERAGE(A1:A20)
the sample standard deviation s with =STDEV.S(A1:A20) (not STDEV.P)
Sample size
n = 20
you chose it
Sample mean
x =
point estimate of μ
Sample std. dev.
s =
our estimate of σ
Is the true mean μ exactly ? Almost certainly not. A different random sample would give a different x. That is why we need a range.
3
The formula when σ is known
Finish Step 2 to unlock.
When the population standard deviation σ is known, we can build a confidence interval for the unknown mean μ using the normal distribution.
In this lab, σ is known. From many years of past applicant data, the admissions office knows that GMAT scores in its pool have a population standard deviation of
σ = 86.80. The population mean μ is still unknown; that is what we estimate.
(You computed s in Step 2, but because σ is known, the formula uses σ.)
Every confidence interval in Chapter 8 has the same skeleton:
point estimate ± critical value × standard error
Margin of error (ME) = critical value × standard error. The interval is point estimate ± ME.
Confidence interval for μ with known σ (slide 20, Eq. 8.6)
CI for μ, σ knownx̄ ± zα/2σ√n
x̄
Sample mean, calculated from a random sample x1, x2, …, xn. This is the point estimate (from your Step 2 sample).
σ
Population standard deviation, assumed known. Here σ = 86.80.
n
Sample size. Here n = 20.
σ/√n
Standard error of x̄: how much sample means typically vary from sample to sample.
1 − α
Confidence level: the middle area under the curve (for example, 0.95).
α
The leftover area, split into two symmetric tails, each with area α/2.
zα/2
Critical z-value: the cutoff that leaves area α/2 in each tail. Determined by the desired confidence level.
Choose a confidence level and watch the picture change:
Middle area = confidence level 1 − αTwo tails, each α/2
Confidence level 1 − α
α
α/2 (each tail)
zα/2
Excel
A higher confidence level gives a larger zα/2 and a wider interval. Greater confidence costs precision.
Your interval, piece by piece
4
Build your 95% confidence interval in Excel
Finish Step 3 to unlock.
Use your sample mean from Step 2, the known σ = 86.80, and the six-step workflow from class. Keep your Excel sheet open next to this page.
Step
Quantity
Excel formula
1
Inputs: x̄, σ, n, confidence level
=AVERAGE(A1:A20), 86.80, 20, 0.95
2
α and critical value zα/2
α = =1−0.95, then =NORM.S.INV(1−α/2)
3
Standard error σ / √n
=86.80/SQRT(20)
4
Margin of error
=z × SE (shortcut: =CONFIDENCE.NORM(α, 86.80, 20))
5
Lower and upper bounds
=x̄ − ME and =x̄ + ME
6
Interpretation
Step 5 on this page
Quantity
Value
Your interval on the GMAT scale
Your 95% CIPoint estimate x̄
5
Interpret your interval
Finish Step 4 to unlock.
Use the reusable sentence frame from class. Here it is with your numbers filled in:
Which statement about your interval is correct?
6
Lift the curtain: did your interval catch μ?
Finish Step 5 to unlock.
This never happens in real life. An analyst never gets to see the true μ. We reveal it here only so you can check how the method performs.
Population size
N = 2,637
True population mean
μ =
fixed; it was there all along
Population std. dev.
σ =
known; used in your interval
The whole applicant pool, with your interval underneath
All 2,637 GMAT scoresTrue μYour CI
Notice that the interval is much narrower than the spread of individual scores. A CI is about the mean score, not about individual applicants.
Your one interval either contains μ or it does not. So where does the "95%" come from? Step 7 answers that.
7
What does "95% confident" mean? Repeat the whole process many times
Finish Step 6 to unlock.
Imagine 100 analysts. Each one draws their own random sample of the same size and builds a confidence interval with the same formula, x̄ ± zα/2σ/√n, exactly as you did. Every interval will be a little different. How many of them will contain the true μ?
Predict first. Out of 100 intervals at the 95% level, about how many will contain μ?
Intervals built
0
Contain μ
0
Miss μ
0
Percent containing μ
–
target: 95%
Each horizontal line is one sample's confidence interval (newest at the top)
Contains μMisses μx̄ of that sampleTrue μ (fixed)
Running percentage of intervals that contain μ
Check the count yourself in Excel
Click Copy all intervals for Excel. Paste into a new sheet at cell A1. You get columns: Sample, x̄, Lower, Upper.
In E1 type Contains μ? In E2 type =AND(C2<=520.78, D2>=520.78) and fill down.
Count the hits: =COUNTIF(E:E, TRUE). Percentage: =COUNTIF(E:E,TRUE)/COUNT(A:A)
Try these experiments
Switch to 99% and draw 100 more. The intervals get wider and fewer of them miss.
Switch to 90%. The intervals get narrower, and about 1 in 10 misses.
Switch to n = 50. The intervals get narrower, but the hit rate stays near the confidence level. Larger samples buy precision, not a different success rate.
Look closely: at a given level and n, every interval has the same width, 2 × zα/2 × σ/√n. Only the center x̄ moves from sample to sample.
Build at least 100 intervals to continue.
8
What did you discover?
Build at least 100 intervals in Step 7 to unlock.
Answer each question on your own first, then check.
1. In the simulation, what moved from sample to sample: μ or the intervals?
The intervals. μ is a fixed number (the dashed line never moved). Each new sample gives a new x̄, so the interval slides around.
2. What does "95% confidence" describe?
The long-run reliability of the procedure. If we drew many samples this way and built one interval from each, about 95% of those intervals would contain μ. Your simulation showed this.
3. Is there a 95% probability that μ is inside your one interval from Step 4?
No. Once your sample is drawn, your interval is fixed and μ is fixed. The interval either contains μ or it does not (probability 1 or 0). We just do not know which, so we say we are "95% confident."
4. What happened when you raised the confidence level from 95% to 99%?
The critical value grew (for example, z from 1.960 to 2.576), so the intervals got wider and fewer missed μ. More confidence costs precision.
5. What happened when you increased n from 20 to 50?
The intervals got narrower, because the standard error σ/√n shrinks as n grows. The percentage that contain μ stayed near the confidence level. A bigger sample gives a more precise estimate at the same confidence.
6. In the simulation, why did all the intervals at the same level and n have the same width?
Because the margin of error is zα/2 × σ/√n, and zα/2, σ and n are the same for every analyst. Only x̄ changes from sample to sample, so each interval is the same-length ruler slid to a different center.
Confidence interval for μ (σ known).x̄ ± zα/2 · σ/√n.
The sample gives the point estimate x̄; the margin of error reflects sampling uncertainty.
Sentence frame: "We are [level]% confident that the true mean GMAT score μ is between [lower] and [upper]. If we drew many samples this way and built one interval from each, about [level]% of those intervals would contain μ."
"The 95% describes the method, not any one interval."