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![Page 1: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/1.jpg)
The Right Questions about Statistics:How confidence intervals work
Maths Learning CentreThe 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.
![Page 2: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/2.jpg)
What is the median number of chapters
in a novel?
Randomly choose 12 books
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 3: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/3.jpg)
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 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 201810 11 13 15 17 19 21 22 23
![Page 4: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/4.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.0005
Under 0.05, so NO, the median is not 10.
Is the median for all books 10?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 5: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/5.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.0005
Under 0.05, so NO, the median is not 11.
Is the median for all books 11?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 6: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/6.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.0005
Under 0.05, so NO, the median is not 12.
Is the median for all books 12?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 7: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/7.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.001
Under 0.05, so NO, the median is not 13.
Is the median for all books 13?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 8: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/8.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.02
Under 0.05, so NO, the median is not 14.
Is the median for all books 14?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 9: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/9.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.23
Over 0.05, so YES, the median could be 15.
Is the median for all books 15?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 10: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/10.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.75
Over 0.05, so YES, the median could be 16.
Is the median for all books 16?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 11: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/11.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 1.0
Over 0.05, so YES, the median could be 17.
Is the median for all books 17?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 12: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/12.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.56
Over 0.05, so YES, the median could be 18.
Is the median for all books 18?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 13: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/13.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.11
Over 0.05, so YES, the median could be 19.
Is the median for all books 19?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 14: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/14.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.01
Under 0.05, so NO, the median is not 20.
Is the median for all books 20?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 15: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/15.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.001
Under 0.05, so NO, the median is not 21.
Is the median for all books 21?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 16: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/16.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.0005
Under 0.05, so NO, the median is not 22.
Is the median for all books 22?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 17: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/17.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
?
SIGN TEST
P-value = 0.0005
Under 0.05, so NO, the median is not 23.
Is the median for all books 23?
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
![Page 18: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/18.jpg)
12 14 16 201810 11 13 15 17 19 21 22 23
That is, all the medians that would give a
p-value of over 0.05.
“95% confidence interval”
It shows all the medians that could be true based on hypothesis tests at the 5% level of significance.
Book # Chapters
1 212 163 144 185 146 167 198 159 17
10 1311 1912 20
This is the (95%)confidence interval
You could say it’s the medians I am “happy to
believe” based on my data.
![Page 19: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/19.jpg)
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
What is the mean weightin a novel?
Randomly choose 12 books
Another example...
![Page 20: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/20.jpg)
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 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) …
Another example...
250 300 350200 400
![Page 21: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/21.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
?
T-TEST
P-value = 0.00007
Under 0.05, so NO, the mean is not 200.
Is the mean for all books 200?
![Page 22: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/22.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
?
T-TEST
P-value = 0.00008
Under 0.05, so NO, the mean is not 201.
Is the mean for all books 201?
![Page 23: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/23.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
?
T-TEST
P-value = 0.00009
Under 0.05, so NO, the mean is not 202.
Is the mean for all books 202?
![Page 24: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/24.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
And so on...
![Page 25: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/25.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
“95% confidence interval”
That is, all the means that would give a
p-value of over 0.05.
It shows all the means that could be true based on
hypothesis tests at the 5% level of significance.
This is the (95%)confidence interval
You could say it’s the means I am “happy to
believe” based on my data.
![Page 26: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/26.jpg)
250 300 350200 400
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
“95% confidence interval”
These ends would have given exactly 0.05 in the
hypothesis test.
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.
So you find them by doing the hypothesis test
calculation backwards...
![Page 27: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/27.jpg)
CI ends = mean ± t* × std. Error= 269 & 347
t = mean – 300 std. Error = 0.46
t* = ± 2.20
More than 0.05, so the mean could be 300.
p = 0.65
HYPOTHESIS TEST
Assume mean is 300
Use assumed mean and data to calculate a test
statistic
Use test statistic to get p-value.
CONFIDENCE INTERVAL
Assume P-value is 0.05
Use p-value to get “critical” test statistics.
Use critical values to get two means.
Compare p-value to 0.05 to decide.
95% confidence interval is between these.
-2.0 -1.0 0.0 1.0 2.0
95% CI is from 269 to 347
250 300 350200 400
![Page 28: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/28.jpg)
Book Weight (g)
1 2422 3663 4244 3125 3076 2387 3178 2659 317
10 31411 21712 379
Let’s go over that again…What is the mean weight in a novel?
Randomly choose 12 books
Choose a hypothesis test, and a
significance level.
T-Test, 0.05
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.
Find the critical values of the test statistic.
t* = ± 2.20
-2.0 -1.0 0.0 1.0 2.0
![Page 29: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/29.jpg)
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.
![Page 30: The Right Questions about Statistics: How confidence intervals work Maths Learning Centre The University of Adelaide A confidence interval is designed.](https://reader038.fdocuments.us/reader038/viewer/2022103112/551c27bd550346a34f8b5ec5/html5/thumbnails/30.jpg)
And this is what aconfidence interval means:
•The values in the CI would be retained with a matching hypothesis test.
•The values in the CI are those you are “happy to believe” based on your data.
•The values in the CI have a high chance of producing data like yours.