For a 99% confidence interval, we want the interval
corresponding to the middle 99% of the normal curve. Z-score for 99%: 2.576
Critical Value for 99%: 0.5% OR .005
For a 95% confidence interval, we want the interval corresponding to the middle 95% of the normal curve
Z-score for 95%: 1.96
Critical Value for 95%: 2.5% 0R .025
For a 90% confidence interval, we want the interval
corresponding to the middle 90% of the normal curve. Z-score for 90%: 1.645
1. There is no sharp border between significant and insignificant, only increasingly strong evidence as the p-value decreases
2. A low p-value doesn’t mean that there is a strong association, only that there is some evidence of some association
Please note: Statisticians will never accept the null hypothesis, we will fail to reject. In other words, we'll say that it isn't or that we don't
have enough evidence to say that it isn't. We will never say that it is, because someone else might come along with another sample which
shows that it isn't and we don't want to be wrong.
If we use the results of a significance test to make a decision, then we either reject the null hypothesis in favor of the alternative hypothesis, or we accept (fail to reject) the null hypothesis. This is
called Acceptance Sampling.
For example, a potato chip manufacturer agrees that each batch of potato chips produced must meet certain quality standards. When a batch of chips is produced, a company employee (Dori Toe) inspects a
sample of the chips. On the basis of the sample outcome, the company either accepts or rejects the entire batch of chips. We will
use acceptance sampling to show how a different concept – inference as decision – changes the reasoning used in tests of
significance.
Errors In Tests Of Significance
When you conduct a test of significance, you're making an assumption, and then deciding whether or not that assumption appears to be true.
We never know if our assumption is true or not—thus, we might be making an error when we reject/fail to reject the null hypothesis.
Going back to the example, let’s say
H0: the batch of potato chips meets standards
Ha: the potato chips do not meet standards
Type I Errors
A Type I Error is the error of rejecting the null hypothesis when in fact it is
true.
(This is known as a false positive.)
Example: We reject a good batch of chips.
The probability of a Type I Error is easy to calculate—if you've
decided that your level of significance is 5%, then you will reject the
null hypothesis if the p-value of the sample mean is less than 5%. How
often should you get sample means with p-values less than 5%?
_______________________________ So the probability of a Type I Error is the
________ as the level of significance — .
Chapter 10 Section 3 & Section 4
Type II Errors
A Type II Error is the error of failing to reject the null hypothesis
when in fact it is False.
(This is known as a false negative.)
Example: We accept a bad batch of chips.
The probability of a Type II Error is more difficult to calculate—it depends on the actual value of μX, which could be anything. Thus, we must calculate the probability of a Type II Error against a specific alternative, μa. To do this, we must first convert
our level of significance into an actual value of . Then, we must find the area to one side of that (the side depends on what kinds of values would lead to rejection). The symbol for the probability of a Type II Error is .
Chapter 10 Section 3 & Section 4
x
x
x
Let's assume that our alternate hypothesis is Ha: . Step 1: Convert (0.05) to .
Chapter 10 Section 3 & Section 4
x
a
Step 2: Assuming that the parameter actually has value , find the probability of failing to reject.
Chapter 10 Section 3 & Section 4
Example #1: In medical disease testing, the null hypothesis is usually the assumption that a person is healthy. The alternative is that he or she has the
disease that we are testing for.
A Type One error is a false positive; a healthy person is diagnosed
with the disease.
A Type Two error is a false negative; an unhealthy person is
diagnosed as disease free.
?
Example #2: A pharmaceutical company claimed on television ads that its new medicine could prevent the common cold. Soon afterward, a consumer
protection group started receiving complaints from consumers that the medicine did not perform as advertised. The consumer protection group decided to
investigate the company’s claim.
H0 :
Ha :
a) Which decision by the consumer protection group might lead to the Type I error?
(b) Which decision by the consumer protection group might lead to the Type II error?
Example #3: Most students enroll in courses with the intention of completing them. But after taking into account their midterm grades,
some students will consider dropping a course. In other words, they want to decide whether to
H0 : Complete the course.
Ha : Drop the course.
(a) Which decision by a student may lead to the Type I error?
Example #4: The manufacturer of a new car claims that it gets 26 miles per gallon (mpg). A consumer group is skeptical of this claim and plans to test H0: = 26 vs. Ha: < 26 at the 5% level of significance
on a sample of 30 cars. Assume that the standard deviation of this kind of measurement is 1.4mpg.
(a) What is the probability of a Type I Error?
Example #4: The manufacturer of a new car claims that
it gets 26 miles per gallon (mpg). A consumer group is
skeptical of this claim and plans to test H
0: = 26 vs. H
a: <
26 at the 5% level of significance on a sample of 30 cars.
Assume that the standard deviation of this kind of
measurement is 1.4mpg.
(b) What is the probability of a Type II Error vs. = 25.8?
First, convert into a value of . Since the alternate hypothesis uses <, we need to reject the null hypothesis only if is too low— below, or to the right, of 26. We need to find that value of that has a left-hand area (away from 26) of 0.05.
Chapter 10 Section 3 & Section 4
a
x
Now we need to find out how often we'll fail to reject (how often will be
greater than ____________) if = 25.8.
Chapter 10 Section 3 & Section 4
There is an _____________ chance of a Type II Error.
Example #5: A company that manufactures classroom chairs for high school students claims that the mean breaking strength of the chairs they make is 300 pounds. From years of production, they have
seen that = 15 pounds. One of the chairs collapsed beneath a 220-pound student last week. You wonder whether the manufacturer is
exaggerating about the breaking strength of their chairs.
(a) State the null and alternative hypotheses in words and symbols.
(b) Describe a Type I error and a Type II error in this situation. Which is more serious?
Example #5: A company that manufactures classroom chairs for high school students claims that the mean breaking strength of the chairs they
make is 300 pounds. From years of production, they have seen that = 15 pounds. One of the chairs collapsed beneath a 220-pound student last week. You wonder whether the manufacturer is exaggerating about the
breaking strength of their chairs.
(c) There are 30 chairs in your classroom. You decide to determine the breaking strength of each chair, and then to find the mean of those values. What values of would cause you to reject H0 at the 5% significance level?
(d) If the truth is that = 270 pounds, find the probability that you will make a Type II error.
Recap of Type I & Type II Errors
It depends on the situation when deciding which type of error is more serious.
*In a jury trial, a Type One error occurs if the jury convicts an
innocent person. A Type Two error occurs if the jury
________________________ a guilty person. Which seems more serious?
*In a Statistics final exam (with H0: the student has learned only 60% of the material), a
Type _____ error would be ____________ a student who in fact learned less than 60% of
the material. A Type _____ error would be ____________ a student who knew enough
to pass. Which seems more serious?
Example #1: A clean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be
tested annually to be sure they meet these standards. Suppose state regulators double check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop’s license if they find significant
evidence that the shop is certifying vehicles that do not meet standards. In this context, what is a Type I error?
In this context, what is a Type II error?
Which type of error would the shop’s owner consider more serious?
Which type of error might environmentalists consider more serious?
Power
When the null hypothesis actually is false, we hope our test is strong enough to reject it. We would like to know how likely we are to
succeed. The _____________ of the test correctly rejects a false null hypothesis. The probability that a hypothesis test will correctly ______________ a __________ null hypothesis is the power of a test.
Definition from text:
The probability that a fixed level significance test will reject H0 when a particular
alternative value of the parameter is true is called the power of the test against that alternative.
The power of a test against any alternative is 1 minus the probability of a
________________ for that alternative. ( )
How can you increase the value of the power in a statistical test of hypotheses?
Chapter 10 Section 3 & Section 4
Free Response Questions
1. An investigator indicates that the power of his test of a sample mean resulting from his research is 0.87. Interpret this statement in your own words.
2. Draw a relatively accurate sketch of a situation dealing with a and a with and .
a. if
b. if
Chapter 10 Section 3 & Section 4
m
0m
aa
=
0.05
m
0<
m
a3. Complete each statement:
a. If the value of increases and n is fixed, then the value of _____________.
b. If n increases, then the power of the test _________________.
c. If n increases, then the values of and ___________________.
