AP Statistics
Name: _____________________
Lesson 9.1 Introduction to Significance Tests
Warmup:
Trace metals found in wells affect the taste of drinking water, and high concentrations can pose a health risk.
Researchers measured the concentration of zinc (in milligrams/liter) near the top and bottom of 10 randomly
selected wells in a large region. The data are listed below:
Well
Bottom
Top
1
0.430
0.415
2
0.266
0.238
3
0.567
0.390
4
0.531
0.410
5
0.707
0.605
6
0.716
0.609
7
0.651
0.632
8
0.589
0.523
9
0.469
0.411
10
0.723
0.612
a) Construct and interpret a 95% Confidence Interval for the mean difference ( µ ) in the zinc
concentrations form these locations in the wells. Use the four step process.
STATE:
PLAN:
DO:
CONCLUDE:
b) If I sampled another two wells and found the difference in the zinc concentration between the top and
bottom of these wells was 0.1039 mg/l and 0.1181. What can I conclude?
AP Statistics
Name: _____________________
Question: Suppose a student claims to be a “Genius.” How would you test to see if it was true?
Significance Tests
A significance test is a formal procedure for comparing observed data with a claim (hypothesis) whose truth we
want to assess. We express the results of a significance test in terms of a probability that measures how well the
data and the claim agree.
Significance testing allows us to use sample data to “test a claim” about a population. For example, testing
whether a population proportion or mean is equal to some value. Examples:
Stating a Hypothesis
A hypothesis is a statement or claim regarding a population. We usually test claims about a specific value of a
population parameter.
Null Hypothesis:
Alternative Hypothesis:
Ex. #1 Does the job satisfaction of assembly-line workers differ when their work is machine-paced rather than
self-paced? One study chose 18 subjects at random from a company with over 200 workers who
assembled electronic devices. Half of the workers were assigned at random to each of two groups. Both
groups did similar assembly work, but one group was allowed to pace themselves while the others
worked on an assembly line that moved at a fixed speed. After two weeks, all of the workers took a test
of job satisfaction. They then switched work settings and after two weeks the test of job satisfaction was
administered again. The response variable is the difference in job satisfaction scores (self-paced minus
machine paced.)
a)
Describe the parameter of interest in this setting.
b)
State appropriate hypothesis for performing a significance test.
AP Statistics
Name: _____________________
Ex. #2 Indicate the null hypothesis and alternative hypothesis for each scenario (both in symbols and words):
a)
A hospital monitors paramedic response times. Last year the average response time was 6.7 minutes
( σ = 2 min). This year a sample of 400 responses had an average of 6.48 minutes. The hospital wants
to know if this is evidence that response times have decreased.
b)
A company claims that their light bulbs will last for 800 hours of use. A consumer wants to know if the
mean lifetime of a bulb is less than 800 hrs.
c)
At Wing’s bar and grill, an order of chicken wings is supposed to weigh 2 pounds. From experience, the
weights of chicken wings order follow a normal distribution with a standard deviation of σ = 0.13 lbs.
You believe that the new chef is making orders that are heavier than 2 lbs.
d)
Pringles advertises that there is only 2 mg of sodium per serving of their chips. A consumer wants to
know if there is any false advertising.
e)
In the past the mean score of Math 12 students in the Ch. 3 test is 67%. This year technology has been
added to the teaching of the unit and the mean score on the test was 72%. Has the use of technology in
the classroom affected the mean score on the Ch. 3 test?
Ex. #3 Here are several situations where there is an incorrect application of hypothesis testing. Explain what is
wrong in each situation.
a)
A change is made to improve student satisfaction with cafeteria food. The null hypothesis, that there is
an improvement, is tested versus the alternative, that there is no change.
b)
A researcher tests the following null hypothesis Ho: x = 20 .
c)
A climatologist wants to test the null hypothesis that it will snow tomorrow.
AP Statistics
Name: _____________________
Note: One-sided alternative hypothesis
Two-sided alternative hypothesis
Defintion: P-value
The probability, computed assuming that Ho is true, that the statistic (such as pˆ or x ) would take a value as
extreme or more extreme than the one actually observed is called the P-value of the test.
P-value small P-value large Ex. #4 Job Satisfaction Study continued. Data collected in the study described in Ex. #1 gave x = 17 and
s X = 60. That is workers rated the self-paced environment, on average, 17 points higher. Researchers
used these data to calculate a P-value of 0.2302.
a)
Explain what the P-value means for the null hypothesis to be true in this setting.
b)
Interpret the P-value in context.
c)
Do the data provide convincing evidence against the null hypothesis? Explain.
Definition: Statistically Significant
If the p-value is smaller than alpha (α ) , we say that the data are statistically significant at the level α . In that
case, we reject the null hypothesis and conclude that there is convincing evidence in favour of the alternative
hypothesis Ho.
If we choose α = 0.05 we are requiring that the data give evidence against Ho so strong that it would happen less
than 5% of the time just by chance when Ho is true.
What does a significance value of 0.01 mean?
AP Statistics
Name: _____________________
Definition: Type I and Type II Errors
If we reject the Ho when Ho is actually true, we have committed a Type I error.
If we fail to reject Ho when Ho is false, we have committed a Type II error.
Ho is true
Ho is false
(Ha is true)
Reject Ho
Fail to reject Ho
Ex. #5 A potato chip producer and its main supplier agree that each shipment of potatoes must meet certain
quality standards. If the producer determines that more than 8% of the potatoes in the shipment have
“blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise,
the entire truckload of potatoes will be used to make potato chips. To make a decision, a supervisor will
inspect a random sample of potatoes from the shipment. The producer will then perform a significance
test using the hypothesis:
Ho: p = 0.08
Ha: p > 0.08
Where p is the actual proportion of potatoes with
blemishes in a given truckload.
Describe a Type I and a Type II error in this setting, and explain the consequences of each.
Coming up ... Probability of Type I and Type II error and how to calculate P-value and test a claim!
Homework: Pg. 547 #1-25 odds, Multiple Choice #27-30