Type I and II errors
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Whenever a decision is made there is the chance
of an error. In statistics decisions are made when
considering hypotheses, either we accept the
null or we reject the null.
If we reject the null, but the null is true, then we
have made a Type I error.
If we accept the null, but the null is false, then
we have made a Type II error.
Type I errors - example 1
Test at the 5% level, whether the sample value of 72 comes from a
normal distribution with a mean of 55 and a variance of 144.
What is the probability of a type I error?
Draw a diagram to show the question
visually:
A type I error is falsely rejecting
the null, so this is simple 5% or
0.05.
Type I errors - example 2
A normal random variable X is
Answer:
described as X~N(80,120).
Draw a diagram to show the
However, it is thought that X has a
question visually:
lower mean and so a sample of size
30 is taken and the following
hypotheses are put forward:
It is decided that if the sample mean
value is less than 76.5 then the null
hypothesis will be rejected.
Find the probability of a Type I
error. -1.75 0
Find the z values. Remember that
this a sample so we have to use the
standard error:
Find the area using tables or a GDC:
p=0.04
Type I errors - example 3 Binomial
A die is suspected of being biased Answer:
towards the six. To test this the die X~N(10,8.333)
is rolled 60 times and two
hypotheses are put forward:
It is decided to reject the null
hypothesis if there are 16 or more
sixes in the 60 rolls of the die.
Find the probability of a type I 0 1.91
error.
A note before starting the solution.
The continuity correction: In this
example we have discrete data Find the area using your calculator
(whole numbers), so we employ the or GDC:
continuity correction. We are going
p=0.028
to start our critical region not at 16
but at 15.5.
Type II errors - example
A normal random variable X is
described as X~N(80,120).
However, it is thought that X has a
lower mean and so a sample of size
30 is taken and the following
hypotheses are put forward:
The probability of a type I error is
5%.
It is later found that the mean is 74,
but the variance remains
unchanged. Calculate the probability
that a type II error is made.
x = 76.7
Firstly a reminder of a type II error:
accepting the null, when it is false. Use the additional information that
the to rewrite the distribution:
Find the critical area and the critical
value. X~N(74,120)
Continued over the page ...
Type II errors - example continued
Draw a new diagram to show the Calculate this area:
new distribution.
Look up this z value to give the
probability.
p = 0.089
The null was accepted for any value
greater than 76.7. Therefore, the
area greater than 76.7 represents
the probability of a type II error.