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Midterm 2 Review: Extra Problems
1
Check Your Understanding
a. (s/False) Using the same random sample from a population, you compute 1)0th a 95% con
fidence interval and a 99% confidence interval for a population proportion. The 99% confidence
interval will always be wider than the 95% confidence interval.
b. (True alse A 90% CI for a proportion is (.442. .542). The sample proportion is .482.
c. (True alse A 90% CI for a proportion is (.442. .542). The margin of error is .10.
d. /Fa se) A 90% (‘I for a proportion is (.142. .542). The 90 percent confidence level means
that if the experiment or study were repeated many times arid we constructed nlany 90 percent
CIsexpect 90 percent of the generated 90% CIs would contain the population proportion.
e False) Every statistic has a sampling distribution
e
f True False) A t-test statistic for a test for one mean tells you how far the sample mean is from
othesized mean value m terms of standard errors of the sample mean
the
g. (True alse A p-value is the probability the null hypothesis is true.
rue a se) The significance leel of a hvpothesis test is the probability of rncori ectly re3ectmg
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the null Ii’ thesis.
i. (True als The power of a test can be increased by either increasing the sample size or de
ignificance level or both.
creasing t
j. (True alse A null hypothesis is rejected only when the p-value is less than the significance
level.
k. False) The p-value for a one-sided test (either direction) can be found based on a twosided -value.
1. Tru False) The test statistic for testing for a difference in two proportions has a normal dis—
n.
trib
Tru False) The test statistic for testing for a difference in tiio proportions involes an esti
m
mate o themmon proportion which is formed by pooling together the two sample proportions.
n. (True/Fal) The standard error of the sample mean estimates roughly how far the observed
observations will differ from the sample mean for repeated samples of the same size.
o. (True/ The test statistic for testing about one population mean has a normal distribution.
1
2
Name That Scenario
One of the primary challenges in hypothesis testing and making confidence intervals is determining
the parameter of interest in a st tidy or experiment. So far. we have covered five scenarios. The
corresponding parameters are: p p p P2 Pd P1 P2
A researcher has decided to investigate attributes of Amherst students for a study on liberal arts
college students. He has identified the following possible questions for his study. hut doesn’t know
the relevant parameters or how to set up his hypotheses. For each, determine the appropriate
parameter and set up hypotheses (there is enough information in each ease to form hypotheses).
—
1. Looking at spending habits, how much money do students spend on average each week on en
tertainment? Is it more than $30 a week on average?
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2. Do males spend more on entertainment than females looking at average spending in a week?
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3. Do more than 40 percent of Amherst students own an Ipod (or other mp3 player)?
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4. In an average weel, how many hours of TV/movies/online “stuff”, etc. do students watch?
Is it different than 10 hours?
10
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5. Does amount of TV/movie/online watching decrease during time at Amherst? Assume you ob
tain a random sample of seniors and they estimate the average number of hours of TV/movies/online
—
“stuff” watched per week during their freshman year and senior ‘ear.
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6. Do upperclassmen and lowerclassmen use facebook at equal rates (assume the question you
ask is: do you use facebook more than 5 times a day?)?
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3
Hypothesis Test or Confidence Interval
Once a parameter has been identified, not every situation calls for a hypothesis test. In fact, some
statisticians argue von should always report a confidence interval, even with hypothesis tests. In
the cases above, the questions were phrased so that hypothesis testing was warranted. However.
CIs are useful estimation tools. Determine the relevant parameter. choose CI or hypothesis test as
most appropriate, and justify your choice for each of the following situations.
1. How much more money
per
week do males spend on food compared
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to
females
on
average?
2. Do more than 60 percent of Amherst students own a personal computer?
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3. Do students get more Than 8 hours of sleep a night o average in their freshman year?
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4. What percentage of Amierst students are internet ad’dicts?
CI
2
4
Review Short Answer
1. What is the general form for a confidence interval?
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standard error of .t as 4.83 for a
2. In a hypothesis test for a population mean, you compute the
sample of 236. Interpret the standard error.
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3. A two-sided t-test for p
the p-value have been if:
would
150 results in a negative test statistic and p-value of .16. What
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than 150?
a. The alternative had been one-sided looking for jz greater
150?
b. The alternative had been one-sided looking for p less than
p
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lp.
normal distributions? (List at least 3 similarities
4. How are t distributions similar or different than
or differences).
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5. Quick answer:
if the null hypothesis is rejected or not if
a. What is the name of the rule used to determine
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comparing the p-value to a significance level?
at a significance level of .03?
test
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hypoth
ed
b. What confidence level is equivalent to a one-sid
conclude that your result was statistically
c. When you do not reject the null hypothesis, can
significant?
t%1o.
riate to use statistical inference (test or CI) to
d. Give an example where it would NOT be approp
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generalize results back to the population.
discussing the t distribution undk?
article
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publish
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e. What pseudo
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Air Pollution Reduction
5
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Data from Moore, McCabe, and Craig
Recall that we have already considered some data from this study. Study details: residents in two
areas had their wheezing symptoms compared after a bypass was constructed in one of the two areas
to remove congestion. The data was collected some time after the bypass was constructed in order
to assess the impact of the reduction in air pollution from having the bypass to remove congestion.
The wheezing data collected was the one-year change in symptoms, i.e. residents reported the
symptoms they had a year previous and how many had improved conditions in the year since was
recorded. When trying to determine if the bypass resulted in a reduction of wheezing symptoms.
one proportion to consider is simply the proportIon of people who reported an improvement
reduction) in the number of wheezing attacks they suffered. In the bypass area. 4 out of 282 people
reported an improvement and for the congested area. 21 out of 163 people reported an improvemerit. For the purposes of this exercise, assume the samples taken were random samples. Is there
significant evidence to conclude that the bypass area had an improvement in number of wheezing
attacks relative to the congested area? Perform a hypothesis test and report your conclusion, as
well as an interpretation of your p-value.
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Fumonisin
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Toxin in Corn (data from FDA)
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Furnonisins are toxins that come from mold and have been found in corn and associated corn prod
ucts. Two different corn meal types had their ftimonisin levels compared. For partially degermed
corn meal, the mean fumonisin level was .59 with a standard deviation of 1.01. For not degermed
corn meal, the mean fumonisin level was 1.21 with a standard deviation of 1.71. Assume the
samples tested of each corn meal type were random samples of size 50.
a. For partially degermed corn meal, assume the population mean fumonisin level is reall .5 with
a population standard deviation of 1. What is the probability of obtaining a sample niean of
for a random sample of size 50? What result was useful in determining this probability?
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b. What is the standard error for the sample mean for partially degermed corn meal? Provide an
interpretation of this standard error.
c. Using the sample data from both the partially degermed and not deg’ermed c n mea, provide
a 95% confidence interval for the difference in population mean amounts of flimonisin in each corn
meal type. Be sure to comment on any assumptions/conditions. (df=79)
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d. Interpret your confidence interval in the context of the problem.
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e. What is the probability that your interval contains the difference in sample means?
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Inositol or Placebo
8
A 1995 article in the American Journal of Psychiatry (Benjamin. J. et al.) described a double-blind
experiment where 21 patients with panic disorder were treated for one week with a placebo and one
week with the drug Inositol. Each patient recorded their number of panic attacks for each week.
The data are listed in Inositol.txt online.
a. What does it mean to say this was a double-blind experiment and why is that important in this
application?
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b. Is the data paired or two independent samples? How can you tell?
c. Perform a preliminary data analysis and describe your findings.
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d. Fhe researchers want to know whether or not Inositol decreased the number of panic attacks
suffered by the patients on average. Determine (and define) an appropriate parameter and set
hypotheses.
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e. What assumptions need to hold in order for the test corresponding to your hypotheses to he
valid? Check your assumptions and comment on their validity.
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f. Perform the appropriate test. What is your test statistic and p value.
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g. Interpret your p-value in the context of the problem.
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ii. Wliat conclusion do you reach about the cfiectiveness of Inositol on decreasing panic attack oc
currences on average compared to placebo? Are there concerns about the validity of the conclusion
based on assumption violations?
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