Drill: Using a table of the normal distribution
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DRAW PICTURES |
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[Otherwise you are guaranteed to get mixed up!] |
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Take X to be normally distributed with
mean = 40 and standard deviation = 8
For these X calculate Z=(X-40)/8 & find Prob(-oo to X) from table.
50 1.25 0.8944
46 0.75 0.7734
34 -0.75 0.2266
32 -1 0.1587 .-----.
Work out the probability that X is | D P |
smaller than 50 0.8944 | R I |
bigger than 50 0.1056 | A C |
bigger than 32 0.8413 | W T |
smaller than 34 0.2266 | U |
bigger than 34 0.7734 | R |
between 34 and 50 0.6677 | E |
between 46 and 50 0.1210 | S |
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What is the average Z? [0] What is the standard deviation of Z? [1]
Where are the quartiles of Z? Q1=-0.6744 Q2=0 Q4=0.6744
Where are the quartiles of X? Q1=34.6 Q2=40 Q3=45.4
BINOMIAL DISTRIBUTION n=8, p=0.5
All data are integer. Each X represents one datum.
0 X 1
1 XXXXXXXX 8
2 XXXXXXXXXXXXXXXXXXXXXXXXXXXX 28
3 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 56
4 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 70
5 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 56
6 XXXXXXXXXXXXXXXXXXXXXXXXXXXX 28
7 XXXXXXXX 8
8 X 1
256 |
This data has a mean of 4 and a standard deviation of 1.42....
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The interval, mean ± one std., is 2.58 to _____________
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What fraction of the data is within that interval?
[Ans: Number of X`s / 256 = 182/256 = 0.71]
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What fraction of the normal distribution is within ± one std.
of the mean? 0.6827
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Compare (2) and (3)
This time the approximation is good, 0.71 is close to 0.68.
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The interval, mean ± two std., is
1.16 to 6.84
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What fraction of the data lies OUSIDE that interval and how does this
compare with the normal approximation?
These are colored red in the diagram.
The fraction is 18/256=0.0703...,
to be compared to the normal 0.04550
All data are integer.
Each X represents one datum.
0 XXXXXXXXXXXXXXXXXXXX 20
1 XXXXXXXXXXXXXXXXXXXX 20
2 XXXXXXXXXXXXXXXXXXXX 20
3 XXXXXXXXXXXXXXXXXXXX 20
4 XXXXXXXXXXXXXXXXXXXX 20
5 XXXXXXXXXXXXXXXXXXXX 20
6 XXXXXXXXXXXXXXXXXXXX 20
7 XXXXXXXXXXXXXXXXXXXX 20
160 |
This data has a mean of 3.5 and a standard deviation of 2.30....
It does not look like the normal distribution. How good or poor is
a normal approximation?