Download 2 Sample Proportions/Means-Mixed Problems File

TWO SAMPLES HT & CI
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AP Statistics
TWO SAMPLES Z for PROPORTIONS: HT & CI
Two Samples Z Test for Proportions
z
 pˆ 1  pˆ 2    p1  p2 
p1 1  p1  p2 1  p2 

n1
z
Two Sample Z Interval for Proportions
 pˆ 1 1  pˆ 1  pˆ 2 1  pˆ 2  



n1
n2


 pˆ 1  pˆ 2   z *
n2
 pˆ 1  pˆ 2    p1  p2 
Assuming unequal variances
1 1
pˆ c 1  pˆ c   
 n1 n2 
x1  x 2
n1  n 2
Assuming equal variances
where pˆ c 
Assumptions:
1. Both random samples are Independent.
2. Sample sizes (n) are less than 10% of population size (N); n < .10N
3. All four products are verified: n1p1  10, n1(1-p1) ≥ 10, n2p2  10 and n2(1-p2) ≥ 10
which indicates both sampling distributions of
p̂
are approx. normally distributed.
Example: Based on its biochemical properties, Finnish researchers hypothesized that
regular use of the sweetner xylitol might be useful for preventing ear infections in
preschool children, and carried out a study to test this hypothesis (Uhari, 1998). In a
randomized experiment, n1 = 165 children took five daily doses of placebo syrup, and 68 of
these children got an ear infection during the three months of the study. Another n2 = 159
children took five daily doses of xylitol, and 46 of these children got an ear infection during
68
 .412 for the
the study. The sample proportions getting an ear infection are pˆ 1 
165
46
 .289 for the xylitol group, and the difference between these
placebo group and pˆ 2 
159
two proportions is .123 (12.3%). Is this observed difference in proportions large enough to
conclude in general that using xylitol reduces the risk of ear infection?
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TWO SAMPLE Z for MEANS: HT & CI
Two Sample Z Test for Means
Two Sample Z Interval for Means
z
 2  2
x1  x2   z * 1  2 
n1
n2


x1  x2   1  2 
 12
n1

 22
n2
Assumptions:
1. Both random samples are Independent.
2. Large enough sample size (n ≥ 30) OR sampled from normal population
x
distributions, so sampling distribution of
will be approximately normally
distributed. (no outliers and little skewness—verify with boxplot)
3. 1 and 2 are known
Note: It is very rare to know both population standard deviations (let alone one), therefore
very rarely will you be expected to construct a CI or perform a HT in this situation.
Fictional Example: A school district wants to investigate if there is a difference in the
mean number of absent days from staff and students. From a national report, it stated that
the population standard deviation of days missed at work by a teacher was 1.6 days and 2.4
days for students. The school district randomly selected 50 teachers and 100 students. The
sample mean for teachers was 3.8 days per school year and 4.7 days per school year for
students. Is there evidence to support this?
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TWO SAMPLE t for MEANS: HT & CI
Two Sample t Test for Means
t
x1  x2   1   2 
2
2
s1
s
 2
n1
n2
Two Sample t Interval for Means
 s2 s 2
x1  x2   t * 1  2 
n
n2
 1

For degrees of freedom (df) either use calculator OR the smaller of n1 – 1 or n2 – 1.
Assumptions:
1. Both random samples are Independent.
2. Large enough sample size (n ≥ 30) OR sampled from approximately normal
x
population distributions, so sampling distribution of
will be approximately
normally distributed. (no outliers and little skewness—verify with boxplot)
3. 1 and 2 are NOT known
Unpooled (separate)
Example: The article “Affective Variables Related to Mathematics Achievement Among
High-Risk College Freshmen” (Psych. Reports (1991):399-403) examines the relationship
between attitudes toward mathematics and success at college-level mathematics. Twenty
men and thirty-eight women selected at random from those identified as being at high risk
of failure (because they did not meet the usual admission requirements for the university)
participated in the study. Each student was asked to respond to a series of questions, and
the answers were combined to obtain a math anxiety score. For this particular scale, the
higher the score, the lower the level of anxiety toward mathematics. Does this data provide
evidence that, as many researchers have hypothesized, the mean anxiety score for women is
different than that for men?
n
s
x
Males
20
35.9
11.9
Females
38
36.6
12.3
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Problems Mixed
1) Much research effort has been expended in studying possible causes of the
pharmacological and behavioral effects resulting from smoking marijuana. The article
“Intravenous Injection in Man of 9THC and 11-OH-9THC” (Science (1982):633) reported
on a study of two chemical substances thought to be instrumental in marijuana’s effects.
Subjects were randomly assigned to one of two treatment groups. Individuals in each group
were given one of the two substances in increasing amounts and asked to say when the
effect was first perceived. The accompanying data values are necessary dose to perception
per kilogram of body weight. Construct a 95% confidence interval and interpret the results.
s
x
9
 THC
19.54
14.47
16.00
24.83
26.39
11.49
18.79
5.91
9
11-OH- THC 15.95
25.89
20.53
15.52
14.18
16.00
18.01
4.42
2) 22071 physicians who were randomly assigned to take aspirin or a placebo every other
day for five years. Of the 11,037 taking aspirin, 104 had a heart attack, while of the 11,034
taking placebo, 189 had a heart attack. Construct a 95% confidence interval to estimate the
difference in the proportion of heart attacks between taking aspirin and placebo.
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3) In an experiment done by social psychologists at the University of California at Berkley.
The researchers either did not stare or did stare at automobile drivers stopped at a campus
stop sign. In the experiment, the response variable was the time (in seconds) it took the
drivers to drive from the stop sign to a mark on the other side of the intersection. The two
populations represented by the observed times are the hypothetical ones that would consist
of the times drivers like these would take to move through a similar intersection, either
under normal conditions (no stare) or the experimental condition (stare). A hypothesis the
researchers wished to test was that the stare would speed up the crossing times, so the
mean crossing time would be greater (slower) for those who did not experience the stare
than it would be for those who did. The mean crossing time was 6.63 seconds with standard
deviation of 1.36 for n = 14 drivers who crossed under normal conditions (no stare) and 5.59
seconds with standard deviation of .822 for n = 13 who crossed in the experimental
condition (stare). The difference between the sample means is 1.04 seconds. Is this
difference large enough to be statistically significant evidence against the null hypothesis?
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4) In an experiment conducted by one of the authors, ten students in a graduate-level
statistics course were given this question about the population of Canada: “The population of
the U.S. is about 270 million. To the nearest million, what do you think is the population of
Canada?” The responses were: 20, 90, 1.5, 100, 132, 150, 130, 40, 200, and 20. Eleven other
students in the same class were given the same question with different introductory
information: “The population of Australia is about 18 million. To the nearest million, what do
you think is the population of Canada?” The responses were: 12, 20, 10, 81, 15, 20, 30, 20, 9,
10, and 20. The experiment was done to demonstrate the anchoring effect, which is that
responses to a survey question may be “anchored” to information provided to introduce the
question. In this experiment, the research hypothesis was that the individuals who saw the
U.S. population figure would generally give higher estimates of Canada’s population than the
individuals who saw the Australia population figure. Is there evidence to support this?
5) An experiment to compare fuel efficiencies for two types of subcompact automobile was
carried out by first randomly selecting n1 = 5 cars of type 1 and n2 = 5 of type 2. Each car
was then driven from Phoenix to Los Angeles by a nonprofessional driver, after which the
fuel efficiency (in mpg) was determined. Is there evidence to support this? The resulting
data with observations in each sample ordered from smallest to largest is given here:
Type I
39.3 41.1 42.4 43.0 44.4
Type II
37.8 39.0 39.8 40.7 42.1
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6) The article “The Sorority Rush Process: Self-Selection, Acceptance Criteria, and the
Effect of Rejection” (J. of College Student Development (1994):346-353) reported on a
study of factors associated with the decision to rush a sorority. Fifty-four women who
rushed a sorority and 51 women who did not were asked how often they drank alcoholic
beverages. For the sorority rush group, the mean was 2.72 drinks per week and the
standard deviation was .86. For the group who did not rush, the mean was 2.11 and the
standard deviation 1.02. Is there evidence to support the claim that those who rush a
sorority drink more than those who do not rush? Test the relevant hypotheses using  =
0.01. What assumptions are required for what test? [Statistic and Data Analysis, Peck,
Olsen, Devore, p.553]
7) Two competing drugs are available for treating math anxiety. There are no apparent side
effects from drug A, whereas there are (nausea and headache) from drug B. A group of
researchers have decided however that they are willing to recommend drug B if the
proportion of cures for the second drug B are higher than the proportion for the first drug
A. The researchers use the drugs experimentally on two groups of freshmen who are
suffering from math anxiety. By the end of the experiment, 52 out of 80 treated with drug
A were classified as cured, whereas 63 out of 90 where were given drug B are so classified.
At the 1% level, should drug B be recommended? (adapted from Michael Legacy)
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