HYPOTHESIS TEST CLASS NOTES
Hypothesis Test: Procedure that allows us to ask a question about an unknown population parameter
Uses sample data to draw a conclusion about the unknown population parameter.
Steps to perform a Hypothesis Test
Step 1: Planning the test: Formulate questions as a pair of hypotheses
AND set criteria for how to draw a conclusion from the data
Step 2: In real life: Gather Data: Select random sample(s) and collect data.
In HW or exam problem: Examine the given data and information, and determine how to
perform test (which distribution and type of test to use)
Step 3: Analyze sample data: Perform a probability calculation to find the “p value”
Step 4: Decide what the data analysis shows about the hypotheses
Step 5: Interpret the decision in the context of the problem.
Step 1: Set up hypotheses that ask a question about the population by setting up two opposite
statements about the possible value of the parameters.
Ho: Null hypothesis: The assumption about the population parameter that will be believed unless
it can be shown to be wrong beyond a reasonable doubt
Ha: Alternate hypothesis:
The claim about the population parameter that must be shown correct
"beyond a reasonable doubt" to believe that it is true.
Statisticians design the hypothesis test so that the outcome that needs to be proved is the alternate hypothesis.
Example A: A new surgery is being tested to determine if is effective in providing a cure.
The hypothesis test should be set up so that the surgery must be proven effective.
Ho: Null hypothesis:
A new surgery is as effective as non-surgical treatment
Ha: Alternate hypothesis:
A new surgery is more effective than non-surgical treatment
A hospital is testing a new surgery for a type of knee injury. Many patients with knee injuries recover
with non-surgical treatment, and surgery has risks. The surgery review board has decided that the
hospital can perform this surgery as a clinical trial.
After studying the medical considerations, they decide that they will approve this type of surgery for
future use if the clinical trial shows that the surgery would cure more than 60% of all such injuries
p = ___________________________________________________________________________
Ho: Null hypothesis:
________________________________________________________________.
Ha: Alternate hypothesis: __________________________________________________________________
Example B: FDA guidelines require that to be considered "fat-free", a serving of salad dressing must
contain less than ½ gram of fat. A salad dressing manufacturer must be able to show that its salad dressing
satisfies these guidelines in order to put the words "fat-free" on its label. (one "serving" is 2 tablespoons of salad
dressing)
The hypothesis test should be set up so that the manufacturer must prove that the salad dressing
satisfies the regulations to call it "fat-free"
µ = ___________________________________________________________________________
Ho: Null hypothesis:
________________________________________________________________.
Ha: Alternate hypothesis: __________________________________________________________________
Page 1
Math 10 RULE FOR HYPOTHESES
Null hypothesis Ho must contain equality of some type: =  or 
Alternate hypothesis Ha must contain a pure inequality.  > or <
Some of the following examples will be used in class. Link to answers is posted on instructor's website.
1. Engineering students at a college want to determine how the price of their calculus book at their bookstore
compares to the price at other college bookstores. The calculus book costs $150 at their bookstore.
They will collect data about the price of this calculus book from 30 other bookstores nationwide and will
use the sample data to decide whether the average price of this book at all other colleges in the nation is
different from the price of $150 at their bookstore.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
2. A soda bottler wants to determine whether the 12 ounce soda cans filled at their plant are underfilled,
containing less than 12 ounces, on average.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
3. Exercise Circuit, a fitness center, advertises a 30 minute workout that rotates clients exercising though
all the fitness stations in 30 minutes. Clients who want longer workouts have complained, but other
clients prefer the relatively quick 30 minute workout. The center conducts a survey to determine
whether the average desired workout time for its clients is longer than the current 30 minute circuit.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
4. It has been estimated that nationally, 16% of US residents lack health insurance coverage.
Suppose that a city wanted to determine whether the percent of city residents without health insurance is
different from the national percent.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
5. The Center for Disease Control reports that only 14% of California adults smoke. A study is
conducted to determine if the percent of De Anza college students who smoke is higher than that.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
6. The average price of a cup of coffee in the airport is at least $2.50
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
7. The average amount of time that students do statistics homework each night is 2 hours.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
8. At most half of all customers at Ace Auto Repair drive foreign cars.
_____ = ____________________________________________________________________________
H0: ________
HA: _________
_________________________
Page 2
PROCEDURE FOR FORMULATING HYPOTHESES: Read problem carefully:
 Is the parameter a MEAN or a PROPORTION?
 Hypotheses always refer to the population parameter p or  (never the sample statistic)
 Describe the parameter, p or  , being tested in a sentence that starts p = description
or  = description
 Write both null hypothesis Ho and alternate hypothesis Ha using mathematical symbolss.
Ho
Null Hypothesis
Contains equality
of some type
≤ or =
 or =
=
Ha
Alternate Hypothesis
Contains strict inequality
only
>
<

One tailed test to the right
One tailed test to the left
Two tailed test
Hypothesis Tests: Correct Decisions and Errors in Decisions
In a hypothesis test, we decide about the hypothesesbased on the strength of evidence in the sample data.
The sample data may lead us to make a correct decision or sometimes make a wrong decision.
We want to make decisions in a way that minimizes the chance of making a wrong decision.
IN REALITY:
Null Hypothesis is TRUE
DECISION:
Based on sample data we
DECIDE NOT TO REJECT null hypothesis
(Data favors null hyothesis)
Based on sample data we
DECIDE TO REJECT null hypothesis
(Data favors alternate hyothesis)
Correct Decision
Type I Error: Wrong Decision
Acceptable probability (risk) is 
 is called significance level
Null Hypothesis is FALSE
Alternate Hypothesis is true
Type II Error: Wrong Decision
Probability is 
Correct Decision
Probability is 1 − 
1 −  is called the power
Type I Error: Rejecting the null hypothesis when in reality the null hypothesis is true
 concluding (based on sample data) in favor of the alternate hypothesis
 when in reality the null hypothesis is true
The probability you are willing to risk of making a Type I error is denoted by  (Greek letter alpha).
 is called the (pre-determined) significance level of the hypothesis test.
The significance level  should be small: a low risk of incorrectly rejecting the null hypothesis if it is really true
Type II Error: Failing to reject the null hypothesis when in reality the null hypothesis is false
 concluding (based on sample data) in favor of the null hypothesis
 when in reality the alternate hypothesis is true
The probability of Type II error is denoted by  (Greek letter beta).
1 −  is called the power of the hypothesis test.
The power of a hypothesis test should be large: large probability of correctly rejecting a false null hypothesis
A Court Trial is a real-life example of a hypothesis test:
Null Hypothesis: Not Guilty: A person is assumed to be innocent
Alternate Hypothesis: Guilty: A person must be proven guilty beyond a reasonable doubt
IN REALITY:
DECISION:
Based on evidence presented in court
jury decides NOT GUILTY
Based on evidence presented in court
jury decides GUILTY
Null Hypothesis is TRUE
Person is NOT GUILTY
Correct Decision
Wrong Decision
Type I Error
Null Hypothesis is FALSE
Person is GUILTY
Wrong Decision
Type II Error
Correct Decision
Page 3
Example A: (see page 1 for the statement of this problem)
p = true population proportion of all people with this knee injury who would be cured if they had this knee surgery
Ho: p  .60
Ha: p > .60
A Type I error would be to conclude that this surgery cures more than 60% of all knee injuries of this
type, when in reality it cures 60% or fewer of all such injuries.
A Type II error would be to conclude that this surgery cures at most 60% of all knee injuries of this type,
when in reality it cures more than 60%of all such injuries.
Some of the examples on page 2 will be used in class to interpret errors; Catalyst website has a link to all answers.
Example #____:
A Type I Error is concluding that_________________________________________________________
when in reality_______________________________________________________________________
A Type II Error is concluding that________________________________________________________
when in reality_______________________________________________________________________
Example #____:
A Type I Error is concluding that_________________________________________________________
when in reality_______________________________________________________________________
A Type II Error is concluding that________________________________________________________
when in reality_______________________________________________________________________
Example #____:
A Type I Error is concluding that_________________________________________________________
when in reality_______________________________________________________________________
A Type II Error is concluding that________________________________________________________
when in reality_______________________________________________________________________
Example #____:
A Type I Error is concluding that_________________________________________________________
when in reality_______________________________________________________________________
A Type II Error is concluding that________________________________________________________
when in reality_______________________________________________________________________






Guidelines:
The interpretation should clearly indicate the conclusion (we conclude, we decide, we believe)
and should indicate what is true in reality
Each part of the interpretation should be stated in context of the problem.
The decision and reality should NOT agree – otherwise it’s a good decision and not an error
Each part of the interpretation should clearly and accurately state a hypothesis (Ho or Ha) in words.
Be careful to reflect equalities and inequalities accurately in your sentences
Page 4
Rare Events
We have an assumption about a property of a population.
We select a sample.

Suppose our sample data has properties that are extremely unlikely to occur if the population actually
has the properties we assumed. We would then conclude that the assumption is not correct.

Suppose our sample data has properties that are reasonably likely to occur if the population actually
has the properties we assumed. This would not give us any reason to doubt the assumption.
Example A: A hospital is testing a new surgery for a type of knee injury The surgery review board has
decided that they will approve this surgery for future use if a clinical trial shows that the true population
cure rate for this surgery would be more than 60%. Otherwise they will not approve it.
Population parameter: p = true population cure rate for this surgery
Random Variable:
P ' = cure rate for a sample of patients having this surgery
Ho: p  .60
Ha: p > .60
Suppose the new surgery is tested on 200 patients.

Suppose the sample proportion of people who are cured is p'=0.90, a 90% cure rate.
What does this seem to indicate about Ho and Ha? Explain:
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________

Suppose the sample proportion of people who are cured is p'=0.46, a 46% cure rate.
What does this seem to indicate about Ho and Ha? Explain:
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________

Suppose the sample proportion of people who are cured is p'=0.605, a 60.5% cure rate.
What does this seem to indicate about Ho and Ha? Explain:
_________________________________________________________________________________________
_________________________________________________________________________________________
_________________________________________________________________________________________
What if p' = 0.62 or 0.65 or 0.70 or 0.75?
Where do we draw the line between "far" and "close"?
 How far away from the null hypothesis should the sample statistic be in order to reject the null
hypothesis and assume the alternate hypothesis is true?
 We use probability calculations to determine where to draw the "dividing line" to decide if a
sample statistic is close to or far from the null hypothesis.
Find the probability of getting a sample that "looks like ours" if the null hypothesis is true.
 If that probability is small, it indicates that our sample is not consistent with the null hypothesis
(that our sample value is considered “far from” the hypothesized value). It would be strong
evidence in support of the alternate hypothesis, leading us to reject the null hypothesis.
 If that probability is large, it indicates that our sample is reasonably likely to occur if the null
hypothesis is true, so does not lead us to reject the null hypothesis
Page 5
Example A: A hospital is testing a new surgery for a type of knee injury The surgery review board has
decided that they will approve this surgery for future use if a clinical trial shows that the true population
cure rate for this surgery would be more than 60%. Otherwise they will not approve it.
Population parameter: p = true population cure rate for this surgery
Random Variable:
P ' = cure rate for a sample of patients having this surgery
Ho: p  .60
Ha: p > .60
Suppose that in a sample of 200 patients having this surgery, 130 of them are cured: p' = 130/200 = 0.65
Is p' = .65 close to or far from the null hypothesis that p = .60 ?
 If sample proportion p' = 0.65 is "close to" (consistent with) the hypothesized proportion 0.60 in Ho,
then we will not be able to conclude that the surgery is effective.
 If sample proportion p' = 0.65 is "far from" (not consistent) the hypothesized proportion 0.60 in Ho,
then we will conclude that the surgery is effective.
Find the probability of getting a sample that "looks like ours" if the null hypothesis is true.
If that probability is small, it indicates that our sample is not consistent with the null hypothesis,
but is strong evidence in support of the alternate hypothesis, leading us to reject the null hypothesis.
Criteria for "what is a small probability?"
The significance level  is our criteria for "what is a small probability?"
It is the risk we are willing to accept of making a Type I error if we reject the null hypothesis.
A Type I Error would be to decide to approve the surgery, believing it's cure rate is more than 60%, (p > .60)
if in reality surgery is not effective, having a cure rate that is not more than 60% ( p ≤ .60)
What risk of making a Type I error are we willing to accept for this situation?_____________
Calculate the probability of getting a sample at least as far from the null hypothesis as our sample is.
This is called the p value. Since our alternate hypothesis says >, we use the right tail of the distribution
If Ho is true, our random variable P ' ~ ____ ( _____ , ___________ )
P(p'≥.65 if p=.60)
P(p'≥.65 | p=.60) = ____________________________________________
is another symbol used for the sample proportion
1-PropZTest on our calculator uses
rather than p'
Compare p value to signficiance level : Is the p value smaller than the significance level? _______
Decision: ______________________________________
Conclusion: ______________________________________________________________________________
______________________________________________________________________________________________
______________________________________________________________________________________________
______________________________________________________________________________________________
DECISION RULE: If p value <  , REJECT Ho
If p value ≥  , DO NOT REJECT Ho
At a (state  as %) level of significance, the sample data DO / DO NOT provide strong
enough evidence to conclude that (state in words what the alternate hypothesis says in context of the problem)
CONCLUSION:
TRY IT: Suppose in a sample of 200 patients having this surgery, 138 of them are cured: p' = 138/200 = 0.69
Does a hypothesis test support adopting the new surgery?
Page 6
Take notes in class on a Chapter 9 Solution Sheet as we do some of these examples in class.
EXAMPLE C: Hypothesis Test of Population Mean  when Population Standard Deviation  is KNOWN
A truck manufacturer company sells Model XDV delivery vans to package delivery services. From
detailed records, the management knows that the average fuel efficiency for these vans is 19.8 miles
per gallon (mpg) with standard deviation 2.9 miles per gallon. Recently they redesigned an engine
part and believe that this redesign will increase fuel efficiency, on average. The engineering team
believes that the known population standard deviation for fuel efficiency of 2.9 miles per gallon will
continue to apply to the new engine design.
A sample of 36 vans with the redesigned engine part has an average fuel efficiency of 21.2 mpg.
20.1
24.6
18.1
15.7
24.8
25.1
24.5
21.5
26.9
13.8
19.9
22.2
20.7
16.6
24.7
24.2
21.7
20.1
22.4
17.7
20.5
22.2
20.5
23.4
24.5
21.1
18.3
18.4
22.2
21.2
21.1
19.9
18.9
19.4
23.1
23.5
At a 5% level of significance, perform a hypothesis test to determine if the redesigned engine part is
effective in increasing the average fuel efficiency.
EXAMPLE D: Hypothesis Test of Population Mean  when Population Standard Deviation  is NOT KNOWN
OptiFiber Network advertises that their fiber optic network delivers internet service at coneection
speeds up to 450 mb per second with an average speed of 300 mb per second. Their main competitor
CableNet claims that internet service from OptiFiber is not advertising truthfully and that their
average connection speed is slower than 300 mb per second. If CableNet can prove that, they will
file a lawsuit against OptiFiber for false and misleading advertising; otherwise their lawyers tell them
they will not be able to win the lawsuit. CableNet selects a sample of 16 different internet
connections on OptiFiber’s network and measures the speed of each connection in the sample.
For the sample, the average speed is 280 mb per second with standard deviation is 64 mb per second.
Does CableNet have sufficient evidence in its sample data to support its claim that OptiFiber’s
average connection speed is less than 300 mb per second? Use a 5% significance level.
228
337
Sample data: internet connection speeds in mb per second n = 16
261
397
200
359
326
224
233
311
211
215
251
367
241
319
(Assume that the underlying population for the speeds of individual internet connections is approximately
normally distributed.)
EXAMPLE E: Hypothesis Test of Population Mean  when Population Standard Deviation  is NOT KNOWN
At Dina's Dress shop, customers complain that fashion designers assume that all women are tall when
they design clothes. Dina read that the designers whose clothes she carries tend to design for women
who are 5 feet 6 inches (66 inches) tall. Because of her customers comments, Dina wants to
determine whether the average height of her customers is shorter.
A random sample of 25 customers has an average height of 64 inches with standard deviation 2.7 inches.
(Assume that the population of individual customer heights is approximately normally distributed.)
Perform a hypothesis test to determine if the average height of all Dina's customers is less than 66
inches. Use a 2% level of significance
NOTE: Then find a confidence interval to estimate the true average height of all Dina's customers,
so that she will be able to decide which designers to buy dresses from for the shop.
Page 7
EXAMPLE F: Hypothesis Test of Population Proportion p
A city government wants to determine if the number of parking spaces required per household for
residentially zoned neighborhoods is still appropriate. Several year ago they based their current
regulations on the assumption that 28% of households are “car-free; now they want to check that
assumption to see if the percent of households that are car free has changed. The conduct a survey of
1200 households; they discover that 372 are “car-free”. At a 3% significance level, can they
conclude that the percent of households that are car free differs from their assumption of 28%?
EXAMPLE G: Hypothesis Test of Population Proportion p
The owners of Ace Auto Repair believe that at most half of all their customers drive foreign cars.
One of their mechanics is taking a Math 10 class and does a hypothesis test as his student project.
He selects a random sample of 50 cars brought in for repair and finds that 21 of them are foreign cars.
At a 5% level of significance, perform the hypothesis test.
EXAMPLE H:
A package for a medication states that this drug contains a dosage of 250 mg per pill.
A lab is quality testing the drug to determine if it contains the correct amount of drug on average.
A sample of 50 pills has an average dosage of 232 mg with standard deviation 25 mg.
At a 2% level of significance, perform the hypothesis test.
(Assume the underlying population of amount of drug in individual pills is approximately normally distributed.)
EXAMPLE I:
The management of SuperSave Grocery Mart is consider adopting Apple Pay. It would require some
investment; they will do this only if they strongly believe that more than 10% of their customers would
use Apple Pay if available.
They survey 300 customers; 42 customers state they would use Apple Pay.
Perform a hypothesis test to determine whether SuperSave Grocery Mart should invest in adopting
Apple Pay.
EXAMPLE J:
The management of SuperSave Grocery Mart is consider adopting Apple Pay. It would require some
investment; they will do this only if they strongly believe that more than 10% of their customers
would
use Apple Pay if available.
They survey 300 customers; 36 customers (12%) state they would use Apple Pay.
Perform a hypothesis test to determine whether the sample data provide sufficient evidence for
SuperSave Grocery Mart to invest in adopting Apple Pay.
Page 8
EXAMPLE K:
Online Shoes claims that the average delivery time for a package with standard shipping is 5 days.
It has recently changed the shipping company it uses and management wants to determine if the
standard shipping time has changed. Shipping records for packages recently shipped with the new
company are examined for a sample of 30 packages.
For the sample of 30 packages, the average shipping time is 5.2 days with standard deviation of 0.6
days. Do the sample data provide sufficient evidence to conclude that the average shipping time for
all packages with the new shipping company is different from 5 days?
(Assume the underlying population of shipping times for individual packages is approximately normally
distributed.)
EXAMPLE L:
Does the label “GLUTEN FREE” guarantee that a food has absolutely NO gluten?
The New York Times on 5/20/2015 published an article titled “Study Finds that Probiotics Labelled
Gluten Free Often Aren’t”. It reported that FDA guidelines state a product must contain less than 20
parts per million of gluten to qualify as gluten free. They reported that some products exceed that but
claim to be gluten free, but also noted that some products containing gluten may legally qualify as
gluten free. Dr. P.H.R. Green of the Celiac Disease Center, Coumbia University, was quoted that
some people “may be much more sensitive to even less than 20 parts per million”.
Suppose that a product claiming to be gluten free was tested and that in a sample of 7 batches of this
product the gluten levels in parts per million were:
9
20
14
22
14
12
15
Perform a hypothesis test to determine if the average gluten content is less than 20 parts per million
so that the manufacturer can support its claim that the product can be represented as gluten free.
Perform a hypothesis test to determine if the average gluten content is less than 20 parts per million
so that the manufacturer can support its claim that the product can be represented as gluten free.
Use a 3% level of significance.
(Assume the underlying population of gluten content in individual batches of in this product is approximately
normally distributed.)
Page 9