Download This chapter is an investigation into an area of statistics known as

Chapter 9: Hypothesis Testing
Section
1
2
3
4
5
Y. Butterworth
Title
Notes Pages
Introduction of Concepts in Ch. 9
2–
Introduction to Statistical Tests
– 11
Testing the Mean μ
12 – 14
Testing a Proportion p
15 – 17
Tests Involving Paired Differences (Dependent Samples) 18 – 19
Testing μ1 – μ2 and p1 – p2 (Independent Samples)
20 – 23
Ch. 9 Brase’s 9th
1
This chapter is an investigation into an area of statistics known as hypothesis testing.
Hypothesis testing is an area of study in inferential statistics. Using hypothesis testing
Statisticians conduct statistical testing to support claims made about populations.
Let's begin with an example before we discuss 2 important concepts in hypothesis testing.
Example:
A manufacturer of a new medication is testing it against an old
medication that has a success rate of 60%. The manufacturer will
not accept the new medication unless it has a success rate better
than the old. If the manufacturer is not very sure about his results
the manufacturer will lose money and possibly even be
investigated for a false claim.
Discussion 1 – Hypotheses
Hypotheses
H0 – The Null Hypothesis (This always contains the equal sign – Brase
will use the equal sign only, but I will use
the opposite inequality with the equal sign
where appropriate.)
Ha or H1 – The Alternative or Research Hypothesis (This is the claim that
we wish to make or
what we wish to
disprove)
Three Types of Hypotheses
H0:  
HA or 1:  >
H0:  
HA or 1:  <
H0:  =
HA or 1:  
One Tail Test (RR in Right or Upper Tail)
One Tail Test (RR in Left or LowerTail)
Two Tail Test (RR in both tails)
Note: The RR stands for rejection region.
Note2: The type of test, left, right or two is determined by the alternative hypothesis.
For Our Example:
We wish to prove that the new medication is better than the old. Since the old had a
60% rate, then the new medication must have more than a 60% rate. This is what we will
make our research hypothesis. Note that this is a proportion, so instead of  we will use
the population proportion, p.
H0: p  60%
(Remember Brase will just use p = 60%)
HA or 1: p > 60%
Y. Butterworth
Ch. 9 Brase’s 9th
2
Discussion 2 – Rejection Region
The rejection region (RR), also called a critical region, is the region in which the
alternative hypothesis is true, the region beyond the critical value. The critical value is
determined with a z or t value based upon the hypothesis  or p.
When the test statistic (the z or t computed based upon the sample data and the value of
) is found to be within the rejection region we reject the null hypothesis and accept
the alternative. This wording is very important!!! We do not prove our alternative we
only accept that it is true. If the test statistic is not in the rejection region, then it leads to
a failure to reject the null hypothesis. Again this wording is very important!!!
Rejection Region
Probability equal to 
Fail to reject H0
Test Stat
t or z
There are three criteria for rejecting.
1) Traditional Method (Critical Value Method) which is based upon comparison of the
test statistic to the z or t value for 
2) P-value Method where the probability of the computed test statistic to the desired .
3) Confidence Interval Method, which we touched upon in 8.4 with the interpretations.
Those interpretations hold for a two-tailed test, and can be modified for a one-tail test.
The traditional method is more straight forward in terms of gauging whether to reject or
fail to reject, but more advanced texts may base their rejection or failure to reject solely
upon p-value. The confidence interval method works very well for two-tail tests but must
be manipulated for a 1-tail test. It also has issues with differences of proportions.
Discussion 3 – Error Discussion
If we reject the null hypothesis when it is true then we are making a serious error!!
For our example, if we reject H0 and accept HA when H0 was actually true, then the
manufacturer could lose money and/or be investigated. Remember, the manufacturer is
wanting to replace a current medication with a new one and will possibly take the old one
off the market if it is shown that the new one will do a much better job!
This is an error that we want to be able to control. This error is called a Type I Error.
We control error by setting the probability of its occurrence. This is our , called the
significance level.
Y. Butterworth
Ch. 9 Brase’s 9th
3
Our Example:
 = 0.01
would be an acceptable probability of
making a Type I error if the outcome of
making such an error can have serious
repercussions.
1   gives the level of confidence in rejecting our null hypothesis.
There is another type of error that can be made as well, it is called a Type II Error. It is
not as serious and can't be controlled, but it is dependent upon  and the population under
HA. The probability of making a Type II Error is symbolized by  (beta). A Type II
Error is a failure to reject the null hypothesis when the null is actually false. The penalty
for such an error in the real world is not as severe as for making a Type I Error.
Our Example:
Type II Error is failing to reject the claim that the
population proportion is less than 60% (the new med isn't better
than the old) when the new medication is better. No
harm is done except that the manufacturer may start
development or trials over again to make a better medicine
– this probably isn't a bad thing – it may cost a little more
money for the consumer and manufacturer in the long run,
but the result will probably be an even better medication!
The following table summarizes error. A similar table can be found on p. 407 of Brase’s
9th edition.
H0 is True
H0 is False
Reject H0
Type I Error

Correct Decision
Fail to Reject H0
Correct Decision
Type II Error

Example:
a)
A corporation maintains a large fleet of company cars for its sales
people. To check the average number of miles driven per month
per car, a random sample of 40 cars is examined. The mean
x-bar = 2752 miles with std. dev., s = 350 miles. Records for the
previous year indicate the average number of miles driven per car
per month was 2600. For  = 0.05 is there a difference between
this year's average mileage and last? Answer the following:
Express the claim that there is no difference in symbols
b)
ID both H0 & HA
Y. Butterworth
Ch. 9 Brase’s 9th
4
c)
ID test as a 2 tail, left or right tailed test
d)
Assume you reject the null hypothesis. State the conclusion.
e)
Assume you fail to reject the null hypothesis. State the conclusion.
Example:
a)
The average live weight of farmer's steers prior to slaughter was
380 lbs. in past years. This year his 50 steers were fed a new diet.
Suppose that they have a mean weight of x-bar = 390 lbs. and
std dev., s = 35.2 lbs. he thinks that his steers weigh more this year
than in the past. Is he correct in his thoughts with an  = 0.01?
Express the claim that there is no difference in symbols
b)
ID both H0 & HA
c)
ID test as a 2 tail, left or right tailed test
d)
Assume you reject the null hypothesis. State the conclusion.
e)
Assume you fail to reject the null hypothesis. State the conclusion.
Example:
a)
The administrator of a nursing home would like to do a time-inmotion study of staff time spent per day performing nonemergency type chores. In particular, she would like to test to see
if efficiency is up (spending less time on chores) from a prior study that
showed that an average of 16 hours were spent on such chores.
Express the claim that there is no difference in symbols
b)
ID both H0 & HA
c)
ID test as a 2 tail, left or right tailed test
d)
Assume you reject the null hypothesis. State the conclusion.
Y. Butterworth
Ch. 9 Brase’s 9th
5
e)
Assume you fail to reject the null hypothesis. State the conclusion.
Discussion 4 – Test Statistics
The test statistics used in this chapter are as follows:
Population Proportions
Population Means
Differences
Z = (p-hat  p0)
( p0 q0 /n)
For  known
Z = (x-bar  0)
/n
For  unknown
T = (x-bar  0)
s/n
Note: The p0, 0 is the value of the null hypothesis!! P-hat, x-bar and s2 come from
sample data used to substantiate your claim. There are 2 methods of looking at this as
previously stated:
Traditional
Test Stat
Vs.
Zcritical
From P(Z>Zcritical)= 
From P(t>tcritical) = 
P-Value
P(Z > Test Stat) = Prob
vs.

Zcritical found with invnormal( or /2
tcritical found with invT  or /2 and n–1
normalcdf(test stat
tcdf(teststat, n-1
When critical value is in critical region
then reject the null & accept alternative
Prob Test Stat <  reject null & accept alt.
Prob Test Stat < /2 reject null & accept alt.
Y. Butterworth
Ch. 9 Brase’s 9th
6
Testing a Claim about a Mean:  Known
Assumptions in this section are the same as those assumptions made for confidence
intervals for populations where  is known.
1)
2)
3)
The sample constitutes a simple random sample
The value of  is known (recall that this rarely happens)
Either the population is known to be Normally distributed or mound
shaped & n>30 or if not mound shaped n > 100. If both are true great!
In this section our test statistic will be the z:
Z =
x-bar  0
 / n
I’ll be doing examples using the traditional method using critical values and the P-value
method using the probability of being beyond the critical value and using the Confidence
Intervals that we developed in the last chapter.
Example:
Recall the example on page 4 that involved company cars. Recall
that x-bar = 2752 and  = 350 & n = 40. At an  = 0.05 test the
claim that last year the average was not the same as last year's
average of 2600. Use the traditional method to test the hypothesis.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
7
Example:
Recall the example involving the live weights of steers where
x-bar= 390,  = 35.2 and n = 50. Test the claim at  = 0.01 that
this year the average weight is more than in past years where the
average weight was 380 lbs. Again use the traditional method.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Example:
Recall the administrator data where x-bar = 12,  = 7.64
and n = 54. For an  = 0.05 test the claim that the
efficiency of the employees has increased (spending less time
on chores) than in the past when the average time on chores
has been 16 hours. Again use the traditional method.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
8
Now let's use the P-Value method to do the company car problem.
P-Value Method
1.
State hypotheses
2.
Draw a diagram and put in value(s) of 
3.
Find the test statistic & find it's probability using your calculator using
the inverse distributions
4.
If the P-value of the test statistic is less than  then reject the null
Example:
Recall the company cars example again. Recall that x-bar = 2752
and  = 350 & n = 40. At an  = 0.05 test the claim that last year
the average was not the same as last year's average of 2600 .
Step1: State the hypotheses
Step2: Draw the picture of
- value(s)
Step3: Find the test statistic & it's probability in z table
Step 4:State the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
9
Now let's do the confidence interval (CI) method on the same data.
CI Method
1.
State the hypotheses
2.
Construct a confidence interval of 1   for a two tailed test and 1  2 for a one
tailed test. CI is about the x-bar not about the H0
3.
If the CI contains the claim in the hypotheses then fail to reject the null
Example:
Let’s use the company cars example again. Recall that
x-bar = 2752 and  = 350 & n = 40. At an  = 0.05 test the claim
that last year the average was not the same as last year's average
of 2600 .
Step1: State the hypotheses
Step2: Construct the confidence interval for a two tailed test
Step 3:State the conclusion (reject if 0 is not in the interval)
Y. Butterworth
Ch. 9 Brase’s 9th
10
Recall the administrator data where x-bar = 12,  = 7.64 and
n = 54. For an  = 0.05 test the claim that the efficiency of the
employees has increased (spending less time on chores) than in the past
when the average time on chores has been 16 hours.
Step1: State the hypotheses
Example:
Step2: Construct the confidence interval for a one-tailed test
(1  2, 90% this time)
Step 3:State the conclusion (reject if 0 is not in the interval)
Let’s use our steer example from before to show how we can use our calculators to do the
tests.
Example:
Recall the example involving the live weights of steers where
x-bar= 390,  = 35.2 and n = 50. Test the claim at  = 0.01 that
this year the average weight is more than in past years where the
average weight was 380 lbs.
a)
Conduct a traditional test.
Use STATTESTSZ-Test and input given information
b)
Conduct a p-value test.
Use the same information from a)
c)
Conduct a CI test.
STATTESTSZ-Interval
This is a 1-tail test so you need a 1 – 2α CI! Recall the interval must be
above the value in the alternative to reject.
Y. Butterworth
Ch. 9 Brase’s 9th
11
§9.2 Testing a Claim About a Mean:  unknown
Assumptions in this section are the same as those assumptions made for confidence
intervals for populations where  is unknown.
1)
2)
3)
The sample constitutes a simple random sample
The value of  is unknown (recall that this is usually the case)
Either the population is known to be normally distributed or
n>30. If both are true great!
In this section our test statistic will be the t:
t =
Example:
x-bar  0
s / n
Measurements of the amounts of suspended solids in
river water on 15 Monday mornings yilds x-bar =47 and
s = 9.4. The water quality is acceptable if the mean
amount of suspended solids is less than 49. Construct an
 = 0.05 test to establish that the quality is acceptable
using a traditional test.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
12
Example:
In a lake pollution study, the concentration of lead in the
upper sedimentary layer of a lake bottom is normally
distributed. In 25 random samples x-bar = 0.38 and s =
0.06. A previous study showed that the mean amount of
lead was 0.34. At an  = 0.01, show that the last study
was incorrect using a p-value test.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
13
Example:
An accounting firm wishes to set a standard time, ,
required for employees to complete a certain audit
operation. The average times from 18 employees yields
x-bar = 4.1 with s = 1.6. The firm believes that it will
take longer than 3.5 hours to complete the audits. Test
their hypotheses at the  = 0.05 level, using a CI test.
Step1: State the hypotheses
Step2: Find the critical value &
draw the picture
Step3: Find the test statistic & locate it on the diagram in 2.
Step 4:State the conclusion
Example:
a)
Y. Butterworth
Redo the last example using your calculator to conduct the 3 types
of tests.
Using the traditional method.
STATTESTST-Test and input statistics
b)
Using the p-value method
Use the same information as a)
c)
Using the CI method
STATTESTST-Interval Input the given information
Remember to adjust for the 1-tail by computing for 1–2α
Ch. 9 Brase’s 9th
14
§9.3 Testing a Claim About a Proportion
Some information that we need to know is that we are that the following
conditions must be met:
1)
Meets the conditions of a binomial distribution (recall ch. 5)
2)
Sample must be large enough and p small enough such that np
> 5 and nq > 5, so that the normal approximation to the
binomial can be assumed
In this section our test statistic will be the z:
Z =
Example:
Step 1:
p-hat  p0
p0q0 / n
A 5 year-old census recorded 20% of families in large
communities lived below poverty level. To determine if this
percentage has changed, a random sample of 400 families is
studied and 70 are found to be living below the poverty level.
Does this finding indicate that the current percentage of families
earning incomes below the poverty level has changed from 5 years
ago? Use  = 0.1 and the traditional method.
State the hypotheses
Step 2:Find the critical values & draw a diagram
 = 0.1 and we want a two tailed test so 0.05 is in each tail
The Z-value that we may have memorized by now from
confidence intervals where there is 0.05 probability above it
is1.645
Step 3:Find the test statistic
Step 4:using the traditional approach compare the Test Stat & Critical
Values. If test stat is in critical region then reject H0 and accept
HA or if it is not in critical region then fail to reject H0.
Y. Butterworth
Ch. 9 Brase’s 9th
15
Example:
A concerned group of citizens wishes to show that less than half of
the voters support the President's handling of a recent crisis. If in a
random sample of 500 voters 228 are in support of the President, at
an  = 0.05 is the claim of the citizens upheld? Use the p-value
method.
Step 1:
State the hypotheses
Step 2:
Calculate the test statistic & draw a picture to find P(z < ztest)
Step 3:
Compare the P(z<ztest) and 
If the value of P(z<ztest) is less than  then reject null
Example:
Y. Butterworth
A buyer from an outing center wants to know if he should carry
more backpacks. He figured if more than ½ of the students at the
local university use backpacks then he could sell more. A random
sample of 78, reveals that 49 carry backpacks. At an  = 0.01, do
you believe he should carry more backpacks? Use the confidence
interval method to compute this probability.
Ch. 9 Brase’s 9th
16
Example:
a)
Y. Butterworth
Use the second example and your calculator to do all 3 methods.
A concerned group of citizens wishes to show that less than half of
the voters support the President's handling of a recent crisis. If in a
random sample of 500 voters 228 are in support of the President, at
an  = 0.05 is the claim of the citizens upheld?
Use the traditional method.
STATTESTS1-PropZTest
Input the correct stats
b)
Use the p-value method
Use the same information as in a)
c)
Use the CI method
STATTESTS1-PropZInt
Input the correct stats
Be sure to correct for the one-tail test by using 1 –2α
Ch. 9 Brase’s 9th
17
§9.4 Tests Involving Paired Differences
This section is about testing the differences between paired observations. This is
dependent data – the data that we mentioned in the discussion in §8.4 but did not discuss
further.
Some Examples of Paired Data:
Height measurements taken for individuals and by the individuals
Paired by individual
IQ’s of identical twins
Paired by twin set
Rainfall in cities across the Bay Area in January and in June
Paired by Bay Area City
Death and Birth Rates for Counties in State
Paired by County
Percentage of males in a wolf pack in the summer and winter for specific
packs
Paired by pack
Number of fish caught on shore vs. in a boat during certain months
Paired by month
Shoe sizes for the left and right foot
Paired by individual
There are many applications of this type of paired data analysis. We need to be able to
establish a clear matching link in order to establish that paired data is to be used. To be
paired data there needs to be a clear, natural way of matching characteristics.
ASSUMPTIONS
1) If normally distributed, then no assumptions are necessary
2) If not able to make a normal assumption, then n≥30
HYPOTHESES
The hypotheses are based upon the differences being positive, negative or zero
H0:
d0 ≥ 0
HA or 1: d0 < 0
H0:
d0 ≤ 0
HA or 1: d0 > 0
H0:
d0 = 0
HA or 1: d0 ≠ 0
CRITICAL VALUE
t α, n – 1
Y. Butterworth
where n= # of paired observations
Ch. 9 Brase’s 9th
18
TEST STATISTIC
t =
*
d  d0
sd /  n
*Note: This value is zero, so the test statistic just looks like: (d-bar•√n)/sd
d-bar is found by computing the difference between the observed values for sample 1 and
sample 2 (or whichever difference is desired) and finding the mean.
sd is the standard deviation of the differences
USING YOUR CALCULATOR
1) Enter sample 1 into register 1 & sample 2 into register 2
2) Go into L3 heading and key in L1 – L3 to get the differences
3) Perform a t-test on the differences in L3
Input: μ = 0, Use Data and choose the correct alternative hypothesis
Output: μ , t-test stat, p-value of test stat (recall if two tail it is twice p-value),dbar (will be notated as x-bar), sd (will be notated as sx)and n
Example:
The following are data for the heights of individuals as reported by the
individual and as measured by a medical professional. The question is at
the 95% Confidence Level is there a difference between the given height
of a individual and the measured height of an individual. (Triola, Elements of
Statistics, 3rd Edition p. 432)
Report
53
Measure 58.1
64
62.7
61
61.1
66
64.8
64
63.2
65
66.4
68
67.6
63
63.5
a)
Give the hypotheses
b)
Give the critical value
c)
Give the test statistic computation
d)
Use your calculator to conduct the test & state the conclusion
Y. Butterworth
Ch. 9 Brase’s 9th
64
66.8
64
63.9
19
§9.5 Testing μ1 – μ2 and p1 – p2 (Independent Samples)
This section is dedicated to differences between independent samples. We will cover,
three types of independent samples: when σ is known for differences between means,
when σ is unknown for differences between means and for differences between
proportions.
Differences between μ1 – μ2 when σ is known
ASSUMPTIONS
1) If normally distributed, then no assumptions are necessary
2) If not able to make a normal assumption, then n1 ≥ 30 & n2 ≥ 30
3) Independent samples
INFORMATION NEEDED
x1
x2
σ1
σ2
HYPOTHESES
The hypotheses are based upon the differences being positive, negative or zero
H0:
μ1 – μ2 ≥ 0
HA or 1: μ1 – μ2 < 0
or
H0:
μ1 ≥ μ 2
HA or 1: μ1 < μ2
H0:
μ1 – μ2 ≤ 0
HA or 1: μ1 – μ2 > 0
or
H0:
μ1 ≤ μ 2
HA or 1: μ1 > μ2
H0:
μ1 – μ2 = 0
HA or 1: μ1 – μ2 ≠ 0
or
H0:
μ1 = μ2
HA or 1: μ1 ≠ μ2
CRITICAL VALUE
z α/2
TEST STATISTIC
z =
x1 – x2 – 0
σ
2
1
/ n1 + σ2 2 / n2
USING YOUR CALCULATOR
1) Enter sample 1 into register 1 & sample 2 into register 2
2) Perform a 2-Sample ZTest on the differences in L3
Input: Use Data(Stats you’ll need x-bars and n’s), σ1,σ2, L1, L2, Freq are 1, and choose
the correct alternative hypothesis
Output: alt. hyp. , z-test stat, p-value of test stat (recall if two tail it is twice p-value),xbars, s’s and n’s
Y. Butterworth
Ch. 9 Brase’s 9th
20
Example:
Let’s do an example from your book, Brase’s Understandable Statistics,
9th edition p. 471 #10.
Based on information from Rocky Mountain News, in a random
sample of 12 winter days in Denver the mean pollution index was
43. Previous studies have shown that σ is 21. A similar study in a
suburb of Denver, called Englewood, a sample of 14 winter days
showed a mean pollution index of 36 and previous studies have
shown that σ is 15. If the pollution indexes for both Denver and
Englewood are assumed to be normally distributed, at the 99%
confidence level test the claim that there is no difference between
the mean pollution index in Denver and Englewood.
Differences between μ1 – μ2 when σ is unknown
ASSUMPTIONS
1) If normally distributed, then no assumptions are necessary
2) If not able to make a normal assumption, then n1 ≥ 30 & n2 ≥ 30
3) Independent samples
4) σ is unknown
INFORMATION NEEDED
x1 , s1 , n1
x2 , s2 , n2
HYPOTHESES
The hypotheses are based upon the differences being positive, negative or zero
H0:
μ1 – μ2 ≥ 0
HA or 1: μ1 – μ2 < 0
or
H0:
μ1 ≥ μ 2
HA or 1: μ1 < μ2
H0:
μ1 – μ2 ≤ 0
HA or 1: μ1 – μ2 > 0
or
H0:
μ1 ≤ μ 2
HA or 1: μ1 > μ2
H0:
μ1 – μ2 = 0
HA or 1: μ1 – μ2 ≠ 0
or
H0:
μ1 = μ2
HA or 1: μ1 ≠ μ2
Y. Butterworth
Ch. 9 Brase’s 9th
21
CRITICAL VALUE
the df is the small of n1 – 1 or n2 – 1
t α, n1 – 1 or n2 – 1
*Note: Statistical software and your TI-Calculator have a more complicated method of computing degrees
of freedom. See exercise #21 for details. Also note that the degrees of freedom for a pooled test is
different as well. DF for pooled is n1 + n2 – 2
TEST STATISTIC
Non-Pooled when no assumptions are made about σ1 & σ2
t =
x1 – x2 – 0
s
2
1
/ n1 + s2 2 / n2
*Note: This is assuming that the population variances are not the same. If the population variances can be
assumed to be the same the standard error (denominator) looks different. The following is for that pooled
instance.
Pooled when σ1 =σ2 is assumed
t =
x1 – x2 – 0
(n1 –
1)s21
+ (n2 – 1)s22 / n1 + n2 – 2
USING YOUR CALCULATOR
1) Enter sample 1 into register 1 & sample 2 into register 2
2) Perform a 2-Sample TTest on the L1 & L2 registers
Input: Use Data(Stats you’ll need x-bars, s’s and n’s), L1, L2, Freq are 1, and choose
the correct alternative hypothesis, for pooled use POOLED Yes &
Non-Pooled use No
Output: alt. hyp. , t-test stat, p-value of test stat (recall if two tail it is twice p-value),df,
x-bars, s’s and n’s
Example:
There are two options for treatment for arthritis. One is a magnet and the
other is considered a “sham” by the manufacturers of the magnets. To test
the claim that the magnets are a better treatment than the “sham,” a study
is done and measurements are taken. For the magnet a sample of 20 yields
a mean of 0.49 and a standard deviation of 0.96 (we aren’t told what these are
measurements of, just that they are indicators of relief and the higher the number the
greater the relief). For the “sham” a sample of 20 yields a mean of 0.44 and
a
standard deviation 1.4.
Y. Butterworth
Ch. 9 Brase’s 9th
22
Differences between p1 – p2
ASSUMPTIONS
1) Independent samples
2) Normal assumptions: np≥5 and nq≥5
INFORMATION NEEDED
p^1 , q^1 , x1 , n1
Note: x1 is # successes, equivalent to p-hat times n
p^2 , q^2 ,x2 , n2
p1 & p2 known, but since they usually aren’t known we need something called the pooled
best estimate for p, or p-bar.
q = 1 – p
p = x1 + x2
n1 + n2
HYPOTHESES
The hypotheses are based upon the differences being positive, negative or zero
H0:
p1 – p2 ≥ 0
HA or 1: p1 – p2 < 0
or
H0:
p1 ≥ p2
HA or 1: p1 < p2
H0:
p1 – p2 ≤ 0
HA or 1: p1 – p2 > 0
or
H0:
p1 ≤ p2
HA or 1: p1 > p2
H0:
p1 – p2 = 0
HA or 1: p1 – p2 ≠ 0
or
H0:
p1 = p2
HA or 1: p1 ≠ p2
CRITICAL VALUE
z α/2
TEST STATISTIC
z =
p^1 – ^p2 – 0
 pq / n1 + pq/ n2
USING YOUR CALCULATOR
1) Perform a 2-Prop ZTest
Input: x1, n1, x2, n2, Alternate hypothesis
Output: alt. hyp. , z-test stat, p-value of test stat (recall if two tail it is twice p-value),
p-hat1, p-hat2 & p-hat which is the pooled p, s’s and n’s
**Recall that the CI method is not technically valid for conducting a hypothesis test.
Example:
Y. Butterworth
A football team passed the ball 247 times in the 2000 season with 83
completed passes. In the 2001 season the same team passed the ball 258
times with 89 completed passes. The coach claims that there is no
difference in pass completion between the two seasons. Test the coach’s
claim at the 90% confidence level.
Ch. 9 Brase’s 9th
23
Y. Butterworth
Ch. 9 Brase’s 9th
24