Grade 7-12 Math Review · Grade 11-12 · Statistics and Probability
Point Estimates and Confidence Intervals
A single number answers the question and hides how sure you are. An interval answers it and says so at the same time.
Find the margin of error
A sample of 100 has mean ₱520; the population σ is ₱80. Build a 95% margin of error.
E = z·σ/√n = 1.96 × 8010 = 1.96 × 8 = ₱15.68, so the interval is ₱504.32 to ₱535.68. C forgets to divide by √n and uses 1.96 × 80. A divides by n as well as taking the root. D is the population SD reported unchanged, which ignores that 100 readings were averaged.
1
A point estimate is one number, and it is almost certainly wrong.
The sample mean x̄ is the best single guess at the population mean μ. It is also
never exactly right — take a different sample and it moves. Reporting it alone tells
the reader nothing about how far it might have moved.
x̄estimates μthe population mean
sestimates σthe population SD
p̂estimates pthe population proportion
The hat and the bar mark the sample quantity; the plain Greek letter is the population one
you cannot see. Keeping them apart on the page is most of what makes this topic readable.
2
The interval is the estimate, plus and minus a margin.
x̄ ± z · σ/√n
Confidence
z
Left outside
90%
1.645
10%, 5% each tail
95%
1.96
5%, 2.5% each tail
99%
2.576
1%, 0.5% each tail
More confidence costs width. A 99% interval is wider than a 95% one on the same data,
because catching μ more often means casting a bigger net. There is no setting that gives
both certainty and precision.
3
A worked interval, in four lines.
σ/√n = 4808 = 60
the standard error
E = 1.96(60) = 117.6
the margin of error
3 200 ± 117.6
centre, then margin
₱3 082.40 to ₱3 317.60
x̄ sits exactly in the middle ✓
The last check is free and worth doing: the sample mean must be the midpoint of the
interval. If it is not, a sign or a subtraction has gone wrong.
4
What “95% confident” actually claims.
It is not a 95% chance that μ is in this interval
μ is a fixed number. It is either inside these limits or it is not — there is no
chance about it once the sample is taken. What is random is the interval: the
procedure produces limits that capture μ in 95% of samples. This one
is either one of the 95% or one of the 5%, and nobody can say which.
A safe wording: “we are 95% confident the mean monthly bill is between ₱3 082
and ₱3 318”. An unsafe one: “there is a 95% probability the mean is
between…”. Examiners mark the difference.
5
Choosing the sample size before you start.
10 = 1.96 × 60√n
set E to what you want
√n = 117.610 = 11.76
rearrange
n = 138.3 → 139
always round UP
Rounding down would leave the margin slightly wider than the 10 that was asked for.
Sample size is the one calculation in this unit where you round up regardless of the
decimal — 138.3 people is 139 people.
6
Variations: back from the limits, other confidence levels, a 90% sample size.
The same x̄ ± z·σ/√n, read in the other direction or with another z.
A 95% interval runs from 46.08 to 53.92. Find x̄, the margin and the standard error.
x̄ = 46.08 + 53.922 = 50
the midpoint
E = 53.92 − 50 = 3.92
half the width
σ/√n = 3.921.96 = 2
undo the z
Given the limits, average them for the centre and halve the gap for the margin.
n = 49, σ = 14, x̄ = 120. Give the 90% and the 99% intervals.
σ/√n = 147 = 2
the same for both
90%: 120 ± 1.645(2) = 120 ± 3.29
116.71 to 123.29
99%: 120 ± 2.576(2) = 120 ± 5.152
114.848 to 125.152
Only z changes; the 99% interval contains the 90% one.
σ = 12. How large a sample gives a 90% margin of error of 2?
2 = 1.645 × 12√n
90% uses 1.645
√n = 19.742 = 9.87
rearrange
n = 97.4 → 98
9.87² ≈ 97.42; round up
Using 1.96 out of habit would give n = 139 here and over-sample by 41 people.
On the exam
Expect a margin of error or an interval from x̄, σ and n, the right z for 90%, 95% or 99%, the correct meaning of “95% confident”, and the sample size for a target margin. Work σ/√n, then E = z·σ/√n, then x̄ ± E; round any sample size up. The 10-second check: x̄ is the exact midpoint, and a higher confidence level gives a wider interval.
Now you try
Practice problem
Build the interval
A sample of 36 students studies a mean of 14 hours a week; the population SD is 3 hours. Give the 95% confidence interval.
σ/√n = 36 = 0.5, so E = 1.96(0.5) = 0.98 and the interval is 13.02 to 14.98 hours. A skips the √n, using 1.96 × 3. B divides by n, 336.D is the z-value written down as if it were the margin, with the standard error left out entirely.
Hint: three separate lines — the standard error σ/√n, then the
margin z·E, then the interval. Skipping straight to the answer is what loses the
√n.
