Worked examples
The question
Question: a sample of 64 has mean 50, and the SD of all the numbers is 8: find the 95% confidence interval for the true mean.
95% interval for μ: x̄ = 50, σ = 8, n = 64
x̄, the sample mean, is 50: the middle of our interval.
σ, the SD of all the numbers, is known: 8.
And n, the sample size, is 64.
x̄: the sample mean, the middle
σ: the SD of all the numbers (known)
n: the sample size
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
interval = x̄ − EBM to x̄ + EBM
1 Find z*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
95% in the middle: 5% left over, 2.5% in each end
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.
left of z* = 0.95 + 0.025 = 0.9750
Now we look for 0.9750 inside the table.
0.9750 is in the table: row 1.9, column .06.
So z = 1.96.
z* = 1.96
2 Error bound
The error bound: z* times σ over the square root of n.
EBM = z* × σ√n
The square root of 64 is 8.
8/8 is 1.
Times 1.96: 1.96.
EBM = 1.96 × 8√64 = 1.96 × 88 = 1.96
Careful!
We divide by the square root of 64, not by 64.
Careful: not 864 ✗
3 The interval
Take the error bound away from the middle: 50 minus 1.96 is 48.04.
= x̄ − EBM = 50 − 1.96 = 48.04
And add it: 50 plus 1.96 is 51.96.
= x̄ + EBM = 50 + 1.96 = 51.96
The 95 percent confidence interval is (48.04, 51.96).
= (48.04, 51.96)
4 Say it in words
In words: we are 95 percent confident that the true mean is between 48.04 and 51.96.
95% confident: 48.04 < μ < 51.96
5 Check
Let's check: the middle of the interval should be 50.
48.04 plus 51.96, divided by 2, is 50.
middle: (48.04 + 51.96) ÷ 2 = 50
Half the width should be the error bound: 51.96 minus 48.04, divided by 2, is 1.96.
So it is right!
half the width: (51.96 − 48.04) ÷ 2 = 1.96
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.
▶ Watch it solved
The question
Question: a sample of 36 has mean 100, and the SD of all the numbers is 15: find the 99% confidence interval for the true mean.
99% interval for μ: x̄ = 100, σ = 15, n = 36
x̄, the sample mean, is 100: the middle of our interval.
σ, the SD of all the numbers, is known: 15.
And n, the sample size, is 36.
x̄: the sample mean, the middle
σ: the SD of all the numbers (known)
n: the sample size
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
interval = x̄ − EBM to x̄ + EBM
1 Find z*
99 percent of the bell is in the middle.
That leaves 1 percent: 0.5 percent in each end.
99% in the middle: 1% left over, 0.5% in each end
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.99 plus 0.005 is 0.9950.
left of z* = 0.99 + 0.005 = 0.9950
Now we look for 0.9950 inside the table.
0.9950 is halfway between 0.9949 and 0.9951.
So z is halfway between 2.57 and 2.58: 2.575.
With more digits it is 2.576.
That is the number books use.
with more digits: z* = 2.576
z* = 2.576
2 Error bound
The error bound: z* times σ over the square root of n.
EBM = z* × σ√n
The square root of 36 is 6.
15/6 is 2.5.
Times 2.576: 6.44.
EBM = 2.576 × 15√36 = 2.576 × 156 = 6.44
Careful!
We divide by the square root of 36, not by 36.
Careful: not 1536 ✗
3 The interval
Take the error bound away from the middle: 100 minus 6.44 is 93.56.
= x̄ − EBM = 100 − 6.44 = 93.56
And add it: 100 plus 6.44 is 106.44.
= x̄ + EBM = 100 + 6.44 = 106.44
The 99 percent confidence interval is (93.56, 106.44).
= (93.56, 106.44)
4 Say it in words
In words: we are 99 percent confident that the true mean is between 93.56 and 106.44.
99% confident: 93.56 < μ < 106.44
5 Check
Let's check: the middle of the interval should be 100.
93.56 plus 106.44, divided by 2, is 100.
middle: (93.56 + 106.44) ÷ 2 = 100
Half the width should be the error bound: 106.44 minus 93.56, divided by 2, is 6.44.
So it is right!
half the width: (106.44 − 93.56) ÷ 2 = 6.44
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.
▶ Watch it solved
The question
Question: a sample of 50 has mean 12.5, and the SD of all the numbers is 3: find the 95% confidence interval for the true mean.
95% interval for μ: x̄ = 12.5, σ = 3, n = 50
x̄, the sample mean, is 12.5: the middle of our interval.
σ, the SD of all the numbers, is known: 3.
And n, the sample size, is 50.
x̄: the sample mean, the middle
σ: the SD of all the numbers (known)
n: the sample size
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
interval = x̄ − EBM to x̄ + EBM
1 Find z*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
95% in the middle: 5% left over, 2.5% in each end
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.
left of z* = 0.95 + 0.025 = 0.9750
Now we look for 0.9750 inside the table.
0.9750 is in the table: row 1.9, column .06.
So z = 1.96.
z* = 1.96
2 Error bound
The error bound: z* times σ over the square root of n.
EBM = z* × σ√n
3 over the square root of 50 is about 0.4243.
Times 1.96: about 0.8316.
EBM = 1.96 × 3√50 = 1.96 × ≈ 0.4243 ≈ 0.8316
Careful!
We divide by the square root of 50, not by 50.
Careful: not 350 ✗
3 The interval
Take the error bound away from the middle: 12.5 minus 0.8316 is 11.6684.
= x̄ − EBM = 12.5 − 0.8316 = 11.6684
And add it: 12.5 plus 0.8316 is 13.3316.
= x̄ + EBM = 12.5 + 0.8316 = 13.3316
The 95 percent confidence interval is (11.6684, 13.3316).
= (11.6684, 13.3316)
4 Say it in words
In words: we are 95 percent confident that the true mean is between 11.6684 and 13.3316.
95% confident: 11.6684 < μ < 13.3316
5 Check
Let's check: the middle of the interval should be 12.5.
11.6684 plus 13.3316, divided by 2, is 12.5.
middle: (11.6684 + 13.3316) ÷ 2 = 12.5
Half the width should be the error bound: 13.3316 minus 11.6684, divided by 2, is 0.8316.
So it is right!
half the width: (13.3316 − 11.6684) ÷ 2 = 0.8316
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.
Answer = (11.6684, 13.3316)
▶ Watch it solved
