Transcript 10.4.1 - GEOCITIES.ws

Type I and Type II Errors
Section 10.4.1
Starter 10.4.1
An agronomist examines the cellulose content of a
variety of alfalfa hay. Suppose that the cellulose
content in the population has a standard deviation
of 8 mg/g. A sample of 15 cuttings has mean
cellulose content of 145 mg/g.
• Give a 90% confidence interval for the mean cellulose
content in the population
• A previous study claimed that the mean cellulose content
was 140 mg/g, but the agronomist believes that the
mean is higher than 140. State Ho and Ha and carry out
a significance test to see if the new data support this
belief.
• Name two major assumptions that underlie the analysis.
Today’s Objectives
• Students should be able to describe the
meaning of Type I and Type II errors in
hypothesis testing.
• Students should be able to compute the
probability of a Type I or a Type II error.
California Standard 18.0
Students determine the P- value for a statistic for a
simple random sample from a normal distribution.
Error Types
• Type I error: Incorrectly reject the null hypothesis when
it is true.
– Example: Convict an innocent person
• Type II error: Incorrectly fail to reject the null hypothesis
when it is false.
– Example: Acquit a guilty person
Ho True
Ho False
Reject Ho
Type I Error
Correct
“Accept” Ho
Correct
Type II Error
Calculating error probabilities
• Type I error: We incorrectly reject Ho
– The significance level (alpha) is the
probability of Type I error.
• Type II error: We should reject Ho
(because it is wrong!) but we incorrectly
fail to do so.
 Use a two step process:
1. First find the range of values that would fail
to reject Ho. (The “acceptance” range)
2. Then find the probability of getting a sample
mean in that range if a certain given value in
Ha is true.
A more detailed look at Type II
• Step 1: Find the values
of  that would lead you
to “accept” Ho
 z* 
x  o
 z*
/ n
 o  z *( / n )  x   o  z *( / n )
For simplicity, let's rewrite that as
• Notice that this looks just a  x  b
like a confidence interval.
• Step 2: Find the
probability that we get an P(a  x  b)  normalcdf (a, b, a , / n )
 within that range if μa is
the true mean
Example
• The mean diameter of a certain type of ball
bearing is supposed to be 2.000 cm. The
diameters are N(2,.010). The customer takes an
SRS of 5 bearings from a shipment and
measures their diameters. He rejects the
bearings if the sample mean diameter is
significantly different from 2.000 at the 5%
significance level.
• State the appropriate hypotheses
 Ho: μ=2
Ha: μ≠2
α = 5%
• State the probability of making a Type I error
– P(Type I error) = 5%
• Find the largest and smallest values of the sample mean
that will lead to the customer accepting the shipment.
(Hint: This is the same as finding a 95% confidence
interval around the assumed mean of 2).
– Stat:Tests:Zinterval(1.9912, 2.0088)
• Now assume that the actual mean diameter of the
shipment is 2.015 cm.
• Based on that assumption, what is the probability that
the sample mean (of the 5 bearings sampled) will lie in
the acceptance range you calculated above?
– Normalcdf(1.9912, 2.0088, 2.015, .00447).0828
• Your result is the probability of a Type II error.
• Suppose the true mean of the shipment is 2.1 cm
– Find the new P(Type II error)
– Normalcdf(1.9913, 2.0088, 2.1, .00447) = 0
• Why is the probability so low?
– The new mean is MANY standard deviations away from the
assumed mean under the null hypothesis. There is virtually NO
CHANCE we will get it wrong.
Today’s Objectives
• Students should be able to describe the
meaning of Type I and Type II errors in
hypothesis testing.
• Students should be able to compute the
probability of a Type I or a Type II error.
California Standard 18.0
Students determine the P- value for a statistic for a
simple random sample from a normal distribution.
Homework
• Read pages 567 - 572
• Do problems 66 - 68