Download Notes 8-2 Comparing Two Proportions

Notes Chapter 22, 24 & 25 BVD
Comparing Two Proportions – Ch 22
When we want to compare the proportions from two groups:
Conditions:
*Both are SRS’s from populations of interest
n1  10% of population 1
n2  10% of population 2
n1p1, n1(1-p1), n2p2, n2(1-p2) are all  10
For Confidence Intervals:
(we do not assume that p1 = p2 therefore we do
not pool for CI's)
( pˆ1  pˆ 2 )  z *
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
Our confidence interval statements now measure “the true difference” in the two
proportions (in context).
For Significance/Hypothesis Testing
Ho: p1 = p2
Ha:
p1  p2 or
p1 < p2 or
p1 > p2
(Since our null states that p1 = p2, we assume this to be true until proven otherwise,
therefore we automatically pool here)
z
pˆ1  pˆ 2
1 1
pˆ (1  pˆ )(  )
n1 n2
Where
pˆ 
x1 x2
n1 n2
We rely on the same p-values and alphas to make our conclusions.
Notes Chapter 24 BVD
Comparing Two Means
Two-Sample t Procedures:
x  x   t  s  s
n n
2
1
1
2
2
2
1
2
*df = there is a nasty formula – ick!
But …calculator gives you this 
t
x x
s s

n n
1
2
2
1
2
2
1
2
Conditions:
*The two groups we are comparing must be independent
of each other.
*Both n1 & n2 must be SRS from the populations of interest
*Both n1 & n2 must be < 10% of their respective
populations of interest
*Sample size restriction must be met:
If n1 + n2 < 15, do not use if outliers or severe
skewness are present
If 15 ≤ n1 + n2 < 30, use except in presence of outliers
If n1 + n2  30, sample is large enough to use
regardless of outliers or skewness by CLT
Pooled Two-Sample t Procedures:
We do not pool t-tests.
Pooling assuming equal σ values. Since σ1 and σ2 are both
unknown, why would we assume they are equal?
“Just say No”.
When we interpret confidence intervals we say “the true difference between two means”
(not the mean difference).
Notes Chapter 25 BVD
Paired Samples
Often we have samples of data that are drawn from populations that are not independent.
We have to watch carefully for those!! We can not treat these samples are two
independent samples, we must consider the fact that they are related.
So…what do we do?
If we have two samples of data that are drawn from populations that are not independent,
we use that data to create a list of differences. That list of differences then becomes our
focus and we will not use the two individual lists again.
We will treat that list of differences as our data.
That list must meet the conditions for a single sample t-distribution. We will use the tinterval or the t-test on this list of differences depending on the question.
When we interpret our confidence interval or make our conclusion we will be talking
about the “mean of the differences.”
Reminders for Single Sample t
Be sure to check conditions for a single sample on the differences.
The One-Sample t-Procedure:
A level C confidence interval for  is:
x  t *(s/ n )
where t* is the critical value from the t distribution based on degrees of freedom.
To test the hypothesis: Ho:  = o and
Ha:  > o
Ha:  < o
Ha:   o
Calculate the test statistics t and the p value.
We make the same conclusions based on p-values and alpha.
For a one-sample t statistic:
t
x
sx
n
has the t distribution with n – 1 degrees of freedom.
Just remember that all the variables represent the “mean difference” for your
populations.
Example 1: Before pharmaceutical drugs are approved for general use, they must
undergo extensive evaluations in clinical trials. During the testing of a drug now used for
the treatment of hypertension, a group of 644 patients received the drug, and 28 reported
headaches as a side effect. A control group of 207 patients receive a placebo and 10
reported headaches as a side effect. For the 2 groups in the study, construct and interpret
a 95 percent confidence interval to estimate the difference in the true proportion of
headaches for each group.
Example 2: After a frost, the owner of 2 orange groves sampled 100 trees from each
grove to estimate the proportion of trees in each grove that had sustained damage. The
sample from the first grove contained 38 damaged trees, while the second grove had 22
damaged trees. Construct a 90 percent confidence interval to estimate the difference in
the proportion of damaged trees in the two groves.
Example 3: I recent study was done to determine whether the antidepressant buproprion
(Zyban) is more effective than the nicotine patch in helping people to quit smoking. A
well designed experiment was conducted with the appropriate randomization and
controls. At the end of the experiment 85 of the 244 Zyban users had quit smoking while
only 52 of the 244 nicotine patch users had quit. Is this good evidence that Zyban is a
better stop smoking aid than the nicotine patch?
Example 4: In a study that was described by a writer for the Washington Post 500
patients undergoing abdominal surgery were randomly assigned to breathe one of two
oxygen mixtures during surgery and for two hours afterward. One group received a
mixture containing 30% oxygen, a standard generally used during surgery. The other
group was given 80% oxygen. Wound infections developed in 28 of the 250 patients who
received 30% oxygen and in 13 of the 250 patients who received 80% oxygen. Is there
sufficient evidence to conclude that the proportion of patients who develop wounds
infections is lower for the 80% oxygen treatment than for the 30% oxygen treatment?
Test at the 5% significance level.
Example 5: The article “Truth or Dare: Tracking Drug Education to Graduation”
compared the drug use of 288 randomly selected high school seniors exposed to a drug
education program (DARE) and 335 randomly selected high school seniors who were not
exposed to such a program. Data for marijuana use are given in the following table:
n
Number who use marijuana
Exposed to DARE
288 141
Not exposed to DARE 335 181
Is there evidence that the proportion using marijuana is lower for students exposed to the
DARE program?
Example 6: Students in a Statistics class at Penn State were asked: “About how many
minutes do you exercise in a week?” For the sixteen women in the class, the mean
number of minutes was 180.63 with a standard deviation of 144.29 minutes. For the
thirteen men in the class, the mean number of minutes was 211.08 with a standard
deviation of 190.85 minutes. Both sets of data are free of outliers and do not contain
severe skewness. Consider these students a simple random sample of all Penn State
students. Is this good evidence that the mean amount of exercise differs for men and
women students at Penn State?
Example 7: Data on the testosterone levels of men in different professions were given in
a paper published in the Journal of Personality and Social Psychology (Dabbs et al.,
1990). The researchers suggested that there are occupational differences in mean
testosterone level. Medical doctors and university professors were two of the
occupational groups studied. The mean testosterone level for 16 medical doctors was
11.60 with a standard deviation of 3.39. The mean testosterone level for 10 university
professors was 10.70 with a standard deviation of 2.59. Give a 95% confidence interval
for the difference in mean testosterone levels of medical doctors and university
professors. Interpret the interval. Can any conclusions be drawn from the interval that
you constructed?
Example 8: Recently a Canadian scientist showed there is a danger that cardboard milk
containers may contain the carcinogen dioxin, which is believed to enter paper products
during the bleaching process. When this occurs it is possible for the dioxin to migrate into
the milk. One solution being considered is to line the cartons with foil. Suppose a paper
manufacturer measures the dioxin in a sample of 50 unlined cartons of milk and obtains a
mean of 0.030 parts per trillion (ppt), with a standard deviation of 0.009 ppt. It also
measures the dioxin in a sample of 50 lined cartons and obtains a mean of 0.006 ppt, with
a standard deviation of 0.002 ppt. Would these results indicate a significant difference in
the mean dioxin levels for the two types of containers?
Example 9: To determine if a new additive improves the mileage performance of
gasoline, seven test runs were conducted with the additive, and six runs were made
without it. The test results are below in miles per gallon (mpg).
With Additive
Without Additive
32.6
31.3
30.1
29.7
29.8
29.1
34.6
30.3
33.5
30.9
29.6
29.9
33.8
Is there sufficient evidence at the 0.05 level to conclude that the additive increases
gasoline mileage?
Example 10: The manager of a large chain retailer believes that camera film sales can be
increased by relocating the film display from the middle to the front of the store. To
investigate this, the manager recorded the number of rolls of film sold for a random
sample of 35 days at the old location and for 30 days at the new location. The results are:
New Location
Old Location
n = 30
n = 35
x-bar = 98.8
x-bar = 95.3
s = 8.9
s = 7.4
Do the results given supply sufficient evidence to support the manager’s claim?
Example 11: Cola makers test new recipes for loss of sweetness during storage. Trained
tasters rate the sweetness before and after storage. (Sweetness is rate on a scale of 1 to 10
with 10 being the sweetest and 1 being the least sweet. )
Before:
After:
8.0
6.0
7.8
7.4
9.3
8.6
6.6
4.6
5.8
6.2
7.8
5.6
8.3
9.6
6.9
5.7
8.7
7.6
7.5
5.2
Is this good evidence that the new cola recipe loses sweetness during storage?
Example 12: Given below are the weights of eight rowers on each of the Cambridge and
Oxford crew teams. Assuming these men represent appropriate random samples, does the
data give good evidence that the average weight on the two teams is different?
Cambridge
188.5
183.0
194.5
185.0
214.0
203.5
186.0
178.5
Oxford
186.0
184.5
204.0
184.5
195.5
202.5
174.0
183.0
Example 13: Ten Pilots performed tasks at a simulated altitude of 25,000 feet. Each pilot
performed the tasks in a completely sober condition and, three days later, after drinking
alcohol. The response variable is the time (in seconds) of useful performance of the tasks
for each condition. The longer a pilot spends on useful performance, the better. The
research hypothesis is that useful performance time decreases with alcohol use, so the
data of interest is the decrease (or increase) in performance with alcohol compared to
when sober.
Pilot #
No Alcohol
Alcohol
1
261
185
2
565
375
3
900
310
4
630
240
5
280
215
6
365
420
7
400
405
8
735
205
9
430
255
10
900
900
Example 14: A weight reduction center advertises that participants in its program lose an
average of at least 5 pounds during the first week of participation. Because of numerous
complaints, the state’s consumer protection agency doubts this claim. To test it, the
records of twelve participants were randomly selected. Their initials weights (in lbs) and
their weights after 1 week in the program appear below. Give a 95% confidence interval
for the mean amount of weight lost in the course of a week. Does this interval support the
weight reduction center’s claim?
Member 1
2
3
4
5
6
7
8
9
10
11
12
Initial
197 153 174 125 149 152 135 143 139 198 215 153
Weight
1 week
195 151 170 123 144 149 131 147 138 192 211 152
later