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yd: 23k; ’@ [ﬁgWm C; 43 639 Chapter 6: Introduction to Inference Here are the Degree of Reading Power (DRP) scores for a sample of 44
third~grade students: www.mm—WW 40 26 39 14 42 18 25 43 46 2? 19
47 19 26 35 34 15 44 40 38 31 46
52 25 35 35 33 29 34 41 49 28 52
47 35 48 22. 33 41 51 27 14 54 45 Suppose that the standard deviation of the population of DR? scores is
known to be cr = 11. Give a 95% conﬁdence interval for the population
mean score. The Conﬁdence Intervals applet lets you simulate large numbers of
conﬁdence intervals quickly. Select 95% conﬁdence and then sample 50
intervals. Record the number of intervals that cover the true value (this
appears in the “hit” box in the applet). Press the reset button and repeat
30 times. Make a sternplot of the results and ﬁnd the mean. Describe the
results. If you repeated this experiment very many times, what would you
expect the average number of hits to be? Repeat the previous exercise [or 80% conﬁdence You are planning a survey of starting salaries for recent liberal arts major
graduates from your college. From a pilot study you estimate that the
standard deviation is about $8000. What sample size do you need to have
a margin of error equal to $500 with 95% conﬁdence? Suppose that in the setting of the previous exercise you are willing to settle _"
for a margin of error of $1000. Will the required sample size be larger or
smaller? Verify your answer by performing the calculations. How large a sample of one~bedroom apartments in Exercise 6.2 would be 
needed to estimate the mean ,u within $3320 with 90% confidence? Researchers planning a study of the reading ability of thirdgrade children
want to obtain a 95% conﬁdence interval for the population mean score 011'
a reading test, with margin of error no greater than 3 points. They carry 1
out a small pilot study to estimate the variability of test scores. The sample standard deviation is s 2 12 points in the pilot study, so in preliminary '
calculations the researchers take the population standard deviation to be .5:
0' I 12. ' {a} The study budget will allow as many as 100 students. Calculate the margin of error of the 95% conﬁdence interval for the population based on n = 100. (3.1:) There are many other demands on the research budget. If all of these
demands were met, there would be funds to measure only 10 child 'ii.
What is the margin of error of the conﬁdence interval based on a = '.'
measurements? . i
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ed on n 6.20 6.21 6.23 Section 6.1 Exercises 433 {c} Find the smallest value of n that would satisfy the goal of a 95%
conﬁdence interval with margin of error 3 or less. Is this sample size
within the limits of the budget? How large a sample of Julie’s potassium levels in Exercise 6.1] would be
needed to estimate her mean ,u within :006 with 95% conﬁdence? In Exercises 6.? and (3.10, we compared conﬁdence intervals based on corn
yields from 35 and 50 small plots of ground. How large a sample is required
to estimate the mean yield within :6 bushels per acre with 90% conﬁdence? To assess the accuracy of a laboratory scale, a standard weight known to
weigh 10 grams is weighed repeatedly. The scale readings are normally
distributed with unknown mean (this mean is 10 grams if the scale has no
bias). The standard deviation of the scale readings is known to be 0.0002 gram. / ta} The weight is weighed ﬁve times. The mean result is 10.0023 grams.
Give a 98% conﬁdence interval for the mean of repeated measurements
of the weight. {b} How many measurements must be averaged to get a margin of error of
i0.0001 with 98% conﬁdence? The Gallup Poll asked 1571 adults what they considered to be the most
serious problem facing the nation’s public schools; 30% said drugs. This
sample percent is an estimate of the percent of all adults who think that
drugs are the schools’ most serious problem. The news article reporting
the poll result adds, “The poll has a margin of errorethe measure of its
statistical accuracy—wotC three percentage points in either direction; aside
from this imprecision inherent in using a sample to represent the whole,
such practical factors as the wording of questions can affect how closely a
poll reflects the opinion of the public in general.” The Gallup Poll uses a complex multistage sample design, but the
sample percent has approximately a normal distribution. Moreover, it is
standard practice to announce the margin of error for a 95% conﬁdence
interval unless a different conﬁdence level is stated. {a} The announced poll result was 30% :r 3%, Can we be certain that the
true population percent falls in this interval? (bl Explain to someone who knows no statistics what the announced result
30% i 3% means. (c) This conﬁdence interval has the same form we have met earlier:
estimate : {caestimate {Actually 0 is estimated from the data, but we ignore this for now.)
What is the standard deviation trauma. of the estimated percent? {at} Does the announced margin of error include errors due to practical
problems such as undercoverage and nonresponse? 434 6.24 69215 6.26 6.27 Chapter 6: Introduction to infereu When the statistic that estimates an unknown parameter has a normal
distribution, a conﬁdence interval for the parameter has the form estimate i 3," new”.
In a complex sample survey design, the appropriate unbiased estimate of the population mean and the standard deviation of this estimate may
require elaborate computations. But when the estimate is known to have
a normal distribution and its standard deviation is given, we can calculate
a conﬁdence interval for it from complex sample designs without knowing
the formulas that led to the numbers given. A report based on the Current Population Survey estimates the 1999
median annual earnings of households as $40,816 and also estimates that
the standard deviation of this estimate is $19]. The Current Population
Survey uses an elaborate multistage sampling design to select a sample
of about 50,000 households. The sampling distribution of the estimated
median income is approximately normal. Give a 95% conﬁdence inteival
for the 1999 median annual earnings of households. The previous problem reports data on the median household income for
the entire United States. In a detailed report based on the same sample survey. you ﬁnd that the estimated median income for fourperson families ' in Michigan is $65,467. Is the margin of error for this estimate with 95% conﬁdence greater or less than the margin of error for the national median,
W'hv? As we prepare to take a sample and compute a 95% conﬁdence interval,
we know that the probability that the interval we compute will cover the
parameter is 0.95. That's the meaning of 95% conﬁdence. If we use several
such intervals, however, our conﬁdence that all give correct results is less
than 95%. Suppose we are interested in conﬁdence intervals for the median
household incomes for three states. We compute a 95% interval for each
of the three, based on independent samples in the three states. (a) What is the probability that all three intervals cover the true median
incomes? This probability (expressed as a percent) is our overall
conﬁdence level for the three simultaneous statements. (b) What is the probability that at least two of the three intervals cover the I true median incomes? The Bowl Championship Series (BCS) was designed to select the top two teams in college football for a ﬁnal championship game. The teams
are selected by a complicated formula. In 2001, the University of Miami
Hurricanes and the University of Nebraska Cornhuskers played for the
championship. However, many football fans thought that Nebraska should
not have played in the game because it was rated only fourth in both major
opinion polls. Thirdranked University of Colorado fans were particularly come for 1:
sample I
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with 95% ﬁnal medi' in tervaL
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for each 6 median
verall tls cover the ._' e top {B teams if Miami
for the
she should both major
Lrticularly 6.28 6.29 6.30 6.2 435 6.2 Tests of Signiﬁcance upset because Colorado soundly beat the Cornhuskers late in the season. A new CNN/USA Todanyallup poll reports that a majority of fans would
prefer a national championship playoff as an alternative to the BCS. The
news media polled a random sample of 1019 adults 18 years of age or older.
A summary of the results states that 54% prefer the playoff, and the margin
of error is 3% for 95% conﬁdence. (a) Give the 95% conﬁdence interval. (b) Do you think that a newspaper headline stating that a majority of fans
prefer a playoff is justiﬁed by the results of this study? Explain your
answer. An advertisement in the student newspaper asks you to consider working
for a telemarketing company. The ad states, "Earn between $500 and $1000
per week.” Do you think that the ad is describing a conﬁdence interval?
Explain your answer. A New York Times poll on women’s issues interviewed 1025 women and
472 men randomly selected from the United States, excluding Alaska and
Hawaii. The poll found that 47% of the women said they do not get enough
time for themselves. (a) The poll announced a margin of error of t3 percentage points for 95%
conﬁdence in conclusions about women. Explain to someone who
knows no statistics why we can't just say that 47% of all adult women
do not get enough time for themselves. (h) Then explain clearly what “95% conﬁdence" means. (c) The margin of error for results concerning men was :4 percentage
points. Why is this larger than the margin of error for women? A radio talk show invites listeners to enter a dispute about a proposed pay
increase for city council members. “What yearly pay do you think council
members should get? Call us with your number.” In all, 958 people call. The
mean pay they suggest is 2? : $9740 per year, and the standard deviation of
the responses is s = $1125. For a large sample such as this, s is very close
to the unknown population a. The station calculates the 95% conﬁdence
interval for the mean pay ,u that all citizens would propose for council
members to be $9669 to $9811. Is this result trustworthy? Explain your
answer. Tests of Significance Conﬁdence intervals are one of the two most common types of formal statisti
cal inference. They are appropriate when our goal is to estimate a population
parameter. The second common type of inference is directed at a quite differ
ent goal: to assess the evidence provided by the data in favor of some claim
about the population. lnferenc : ally Ho ‘ _
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sample of : 6.32 6.33 63.34 Section 6.2 Exercises 453 In each of the following situations, a signiﬁcance test for a population mean
u is called for. State the null hypothesis HO and the alternative hypothesis
HCl in each case. {a} Experiments on learning in animals sometimes measure how long it
takes a mouse to ﬁnd its way through a maze. The mean time is 18
seconds for one particular maze. A researcher thinks that a loud noise
will cause the mice to complete the maze faster. She measures how
long each of 10 mice takes with a noise as stimulus. {it} The examinations in a large history class are scaled after grading so
that the mean score is 50. A self—conﬁdent teaching assistant thinks that
his students have a higher mean score than the class as a whole. His
students this semester can be considered a sample from the population
of all students he might teach, so he compares their mean score with
50. {c} The Census Bureau reports that households spend an average of
31% of their total spending on housing. A homebuilders association
in Cleveland wonders if the national ﬁnding applies in their
area. They interview a sample of 40 households in the Cleveland
metropolitan area to learn what percent of their spending goes toward
housing. In each of the following situations, state an appropriate null hypothesis
H0 and alternative hypothesis Hg. Be sure to identify the parameters that
you use to state the hypotheses. (We have not yet learned how to test these
hypotheses.) (a) A sociologist asks a large sample of high school students which
academic subject they like best. She suspects that a higher percent
of males than of females will name mathematics as their favorite
subject. {is} An education researcher randomly divides sixthgrade students into
two groups for physical education class. He teaches both groups
basketball skills, using the same methods of instruction in both classes.
He encourages Group A with compliments and other positive behavior
but acts cool and neutral toward Group B. He hopes to show that
positive teacher attitudes result in a higher mean score on a test of
basketball skills than do neutral attitudes. (e) An economist believes that among employed young adults there is a
positive correlation between income and the percent of disposable
income that is saved. To test this, she gathers income and savings data
from a sample of employed persons in her city aged 25 to 34. Translate each of the Following research questions into appropriate H0
and Hg. (23.) Census Bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research ﬁrm
questions shoppers at the mall to ﬁnd out whether the mean household
income of mall shoppers is higher than that of the general population. vInfE .:',} : of
5 who
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iis conclus‘ 6.4:} oil21 6.43 6.44 6.45 Section 6.2 Exercises 455 especially the P~value, as if you were speaking to a doctor who knows no
statistics. A social psychologist reports that “ethnocentrisrn was signiﬁcantly higher
(P < 0.05) among church attenders than among nonattenders." Explain
what this means in language understandable to someone who knows no
statistics. Do not use the word "signiﬁcance" in your answer. A study examined the effect of exercise on how students perform on their
ﬁnal exam in statistics. The P—value was given as 0.87. (a) State null and alternative hypotheses that could have been used for this
study. (Note that there is more than one correct answer.) (b) Do you reject the null hypothesis?
(c) What is your conclusion? {d} What other facts about the study would you like to know for a proper
interpretation of the results? The ﬁnancial aid ofﬁce of a university asks a sample of students about
their employment and earnings. The report says that "for academic year
earnings, a signiﬁcant difference ( P 2 0.038) was found between the sexes,
with men earning more on the average. No difference (P r 0.476) was
found between the earnings of black and white students.”5 Explain both of
these conclusions, for the effects of sex and of race on mean earnings, in
language understandable to someone who knows no statistics. Statistics can help decide the authorship of literary works. Sonnets by a
certain Elizabethan poet are known to contain an average of ,u 2 6.9 new
words (words not used in the poet's other works). The standard deviation of
the number of new words is o = 2.7. Now a manuscript with 5 new sonnets
has come to light, and scholars are debating whether it is the poet's work.
The new sonnets contain an average off : 1 1.2 words not used in the
poet’s known works. We expect poems by another author to contain more
new words, so to see if we have evidence that the new sonnets are not by
our poet we test H0: ,LL 1: 6.9
Ha: ,u > 6.9 Give the z test statistic and its P~value. What do you conclude about the
authorship of the new poems? The Survey of Study Habits and Attitudes (SSHA) is a psychological test
that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. The mean score for U.S. college
audenmisabout115,andthesunuwrddewanonisabout30.Ateachm"who
suspects that older students have better attitudes toward school gives the
SSHA to 20 students who are at least 30 years of age. Their mean score is 35 = 135.2. 468 6.74 6.75 6.76 6.77 Chapter 6: Introduction to free of colds. This difference is statistically signiﬁcant (P =2 0.03) in fat '
the vitamin C group. Can we conclude that vitamin C has a strong client
preventing colds? Explain your answer. Every user of statistics should understand the distinction between stati
signiﬁcance and practical importance. A sufﬁciently large sample will
declare very small effects statistically signiﬁcant. Let us suppose that S mathematics (SATM) scores in the absence of coaching vary normally ‘
mean pi = 475 and 0' = 100. Suppose further that coaching may cha .t
but does not change on An increase in the SATM score from 475 to 478 '
no importance in seeking admission to college, but this unimportant :;E
can be statistically veiy signiﬁcant. To see this, calculate the Pvalue for i.
test of H0: it : 475
HQ: ,u > 475 in each of the following situations: {a} A coaching service coaches 100 students; their SATM scores avera
I“ : 478. .
(b) By the next year, the service has coached 1000 students; their SATM
scores average a? z 478. (c) An advertising campaign brings the number of students coached to
10,000; their average score is still I r 478. Give a 99% conﬁdence interval for the mean SATM score it after coach):
in each part of the previous exercise. For large samples, the conﬁdence ‘
interval says, "Yes, the mean score is higher after coaching, but only by 5..
small amount.” I a As in the previous exercises, suppose that SATM scores vary normally i;
or : 100. One hundred students go through a rigorous training program.
designed to raise their SATM scores by improving their mathematics Carry out a test of H0? it : 475
Ha: ,u > 475 in each of the following situations:
{a} The students’ average score is 2? 2 491.4. Is this result signiﬁcant at: 5% level? :
{b} The average score is f : 491.5. Is this result signiﬁcant at the 5%! The difference between the two outcomes in (a) and (b) is of no i :
tance. Beware attempts to treat a: : 0.05 as sacred. A local television station announces a question for a call~in opinion poll g.
the six o’clock news and then gives the response on the eleven o’clock 11
Today’s question concerns a proposed increase in funds for student lea ~
Of the 2372 calls received, 1921 oppose the increase. The station, follow ...
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This note was uploaded on 11/20/2009 for the course ECON 5000 taught by Professor Smith during the Summer '09 term at York University.
 Summer '09
 Smith

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