1. How confidence intervals work
The Right Questions about Statistics:
How confidence intervals work
Maths Learning Centre
The University of Adelaide
A confidence interval is designed to give a RANGE of possible answers
for a “WHAT’S THE NUMBER?” question, using DATA from a sample.
You calculate which numbers could have been likely to produce your data.

2. What is the median number of chapters in a novel?
Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
What is the median number of chapters in a novel?
Randomly choose 12 books

3. Which possible medians might be likely to produce this data?
Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Which possible medians might be likely to produce this data?
For each particular median, we could do a hypothesis test to answer the question (in this case it’s called the sign test) … 12 14 16 20 18 10 11 13 15 17 19 21 22 23

4. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 10?
SIGN TEST
P-value =
Under 0.05, so NO, the median is not 10. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

5. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 11?
SIGN TEST
P-value =
Under 0.05, so NO, the median is not 11. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

6. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 12?
SIGN TEST
P-value =
Under 0.05, so NO, the median is not 12. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

7. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 13?
SIGN TEST
P-value = 0.001
Under 0.05, so NO, the median is not 13. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

8. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 14?
SIGN TEST
P-value = 0.02
Under 0.05, so NO, the median is not 14. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

9. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 15?
SIGN TEST
P-value = 0.23
Over 0.05, so YES, the median could be 15. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

10. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 16?
SIGN TEST
P-value = 0.75
Over 0.05, so YES, the median could be 16. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

11. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 17?
SIGN TEST
P-value = 1.0
Over 0.05, so YES, the median could be 17. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

12. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 18?
SIGN TEST
P-value = 0.56
Over 0.05, so YES, the median could be 18. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

13. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 19?
SIGN TEST
P-value = 0.11
Over 0.05, so YES, the median could be 19. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

14. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 20?
SIGN TEST
P-value = 0.01
Under 0.05, so NO, the median is not 20. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

15. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 21?
SIGN TEST
P-value = 0.001
Under 0.05, so NO, the median is not 21. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

16. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 22?
SIGN TEST
P-value =
Under 0.05, so NO, the median is not 22. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

17. Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
Is the median for all books 23?
SIGN TEST
P-value =
Under 0.05, so NO, the median is not 23. ? 12 14 16 20 18 10 11 13 15 17 19 21 22 23

18. This is the (95%) confidence interval
Book
# Chapters 1 21 2 16 3 14 4 18 5 6 7 19 8 15 9 17 10 13 11 12 20
This is the (95%) confidence interval
It shows all the medians that could be true based on hypothesis tests at the 5% level of significance.
That is, all the medians that would give a p-value of over 0.05.
You could say it’s the medians I am “happy to believe” based on my data.
“95% confidence interval” 12 14 16 20 18 10 11 13 15 17 19 21 22 23

19. What is the mean weight in a novel?
Another example...
Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
What is the mean weight in a novel?
Randomly choose 12 books

20. Which possible means might be likely to produce this data?
Another example...
Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
Which possible means might be likely to produce this data?
For each particular mean, we could do a hypothesis test to answer the question (in this case it’s called the t-test) …
200
250
300
350
400

21. Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
Is the mean for all books 200?
T-TEST
P-value =
Under 0.05, so NO, the mean is not 200. ?
200
250
300
350
400

22. Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
Is the mean for all books 201?
T-TEST
P-value =
Under 0.05, so NO, the mean is not 201. ?
200
250
300
350
400

23. Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
Is the mean for all books 202?
T-TEST
P-value =
Under 0.05, so NO, the mean is not 202. ?
200
250
300
350
400

24. Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
And so on...
200
250
300
350
400

25. This is the (95%) confidence interval
Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
This is the (95%) confidence interval
It shows all the means that could be true based on hypothesis tests at the 5% level of significance.
That is, all the means that would give a p-value of over 0.05.
You could say it’s the means I am “happy to believe” based on my data.
“95% confidence interval”
200
250
300
350
400

26. You just have to figure out where the ends are.
Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
BUT you don’t have to go through all this to find a confidence interval!
You just have to figure out where the ends are.
These ends would have given exactly 0.05 in the hypothesis test.
So you find them by doing the hypothesis test calculation backwards...
“95% confidence interval”
250
300
350
200
400

27. HYPOTHESIS TEST CONFIDENCE INTERVAL
95% CI is from 269 to 347
95% confidence interval is between these.
Assume mean is 300
Use assumed mean and data to calculate a test statistic
t = mean – 300
std. Error
= 0.46
CI ends = mean ± t* × std. Error
= 269 & 347
Use critical values to get two means.
Use test statistic to get p-value.
t* = ± 2.20
p = 0.65
Use p-value to get “critical” test statistics.
Compare p-value to 0.05 to decide.
More than 0.05, so the mean could be 300.
Assume P-value is 0.05
CONFIDENCE INTERVAL
250
300
350
200
400
-2.0
-1.0
0.0
1.0
2.0

28. Let’s go over that again…
Book
Weight (g) 1
242 2
366 3
424 4
312 5
307 6
238 7
317 8
265 9 10
314 11
217 12
379
What is the mean weight in a novel?
Randomly choose 12 books
Choose a hypothesis test, and a significance level.
T-Test, 0.05
Find the critical values of the test statistic.
t* = ± 2.20
Use critical values and your data to calculate the two ends.
CI ends = mean ± t* × std. Error
= 269 & 347
State the answer.
The 95% CI is from 269g to 347g.
-2.0
-1.0
0.0
1.0
2.0

29. So this is how to find a confidence interval:
Have a “what’s the number?” question.
Collect data.
Choose a matching hypothesis test.
Work backwards to calculate two ends.
The confidence interval is between these two values.

30. And this is what a confidence interval means:
The numbers in the CI would have been retained with a matching hypothesis test asking, “Is this number equal to this value?”
The values in the CI have a high chance of producing data like yours.
The values in the CI appear to be consistent with your data.
OUTSIDE
NOT CONSISTENT WITH DATA
INSIDE
APPEARS CONSISTENT WITH DATA