Chapter 9 Estimating Using a Single Sample
9.1 Point Estimation
Point Estimate
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The paper “U.S. College Students’ Internet Use: Race, Gender and Digital Divides” (Journal of
Computer-Mediated Communication, 2009) reports the results of 7421 students at 40 colleges and
universities. (The sample was selected in such a way that it is representative of the population of
college students.)
The authors want to estimate the proportion (p) of college students who spend more than 3 hours a
day on the Internet. 2998 out of 7421 students reported using the Internet more than 3 hours a day.
𝑝̂ =
The paper “The Impact of Internet and Television Use on the Reading Habits and Practices of
College Students” (Journal of Adolescence and Adult Literacy, 2009) investigates the reading habits
of college students. The following observations represent the number of hours spent on academic
reading in 1 week by 20 college students.
1.7
3.8
4.7
9.6
11.7
12.3
12.3
12.4
12.6
13.4
14.1
14.2
15.8
15.9
18.7
19.4
21.2
21.9
23.3
28.2
sample mean 𝑥̅ =
sample median=
10% trimmed mean=
Choosing a Statistic for Computing an Estimate
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Homework: 1-8
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9.2 Large-Sample Confidence Interval Population Proportion
Confidence Intervals
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Confidence Level
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General Properties for Sample Distributions of 𝑝̂
1.
2.
3.
Developing a Confidence Interval
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The Large-Sample Confidence Interval for p
The general formula for a confidence interval for a population proportion p when
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The article “How Well Are U.S Colleges Run?” (USA Today, February 17, 2010) describes a survey
of 1031 adult Americans. The survey was carried out by the National Center for Public Policy and
the sample was selected in a way that makes it reasonable to regard the sample as representative of
adult Americans. Of those surveyed, 567 indicated that they believe a college education is essential
for success.
What is a 95% confidence interval for the population proportion of adult Americans who believe that
a college education is essential for success?
Verify Conditions:
Calculation:
Conclusion:
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Choosing a Sample Size
The bound on error estimation for a 95% confidence interval is
Solved for n:
What do we use for p?
In spite of the potential safety hazards, some people would like to have an internet connection in
their car. Determine the sample size required to estimate the proportion of adult Americans who
would like an internet connection in their car to within 0.03 with 95% confidence.
1. A. What is the z critical value for a 95% CI?
B. For a 99% CI?
C. For a 90% CI?
2. If we increase the confidence level, what happens to the CI?
3. If we obtain a larger sample, what happens to the CI?
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4. The city council in a large city has become concerned about the trend toward exclusion of renters
with children in apartments in the city. To determine the extent of this problem, they selected a
random sample of apartments and determined for each whether or not children would be permitted.
Sample Results: 37 of the 250 sampled apartments exclude renters with children.
Goal: It is desired to estimate the proportion of all apartments in the city that exclude renters with
children with 95% confidence.
A. Determine the population type (numerical or categorical) and describe the population
characteristic in words.
B. How much confidence in desired? Write the formula for the confidence interval.
C. Check that the sample meets the necessary assumptions and then determine n and 𝑝̂ .
D. Construct the CI. Show your work.
E. Make a concluding statement interpreting your CI.
F. The housing coordinator claims that the proportion of apartments in the city that prohibit
children is greater than .10. Does the CI support that claim? Explain.
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5. The Rippin’ Good Company makes chocolate chip cookies that are less expensive than namebrand cookies. Are their Chunky Chips-A-Plenty as good as Nabisco’s Chunky Chips-Ahoy? To
answer that question, a random sample of college students is given a blind taste test of Chunky
Chips-A-Plenty cookies and Chunky Chips-Ahoy cookies. 18 out of the 44 students preferred
Chunky Chips-A-Plenty. Construct an estimate, with 95% confidence, of the proportion of all college
students who would choose Chunky Chips-A-Plenty over Chunky Chips-Ahoy. Show all necessary
information, including verifying that your method is appropriate.
a. Based on your CI result, is it possible that at least 50% of all college students prefer Chunky
Chips-A-Plenty over Chunky Chips-Ahoy? Explain.
b. Based on your CI result, is it possible that at least 50% of all consumers prefer Chunky Chips-APlenty over Chunky Chips-Ahoy? Explain.
A confidence interval is computed by: Point Estimate ± “Margin” of Error
=(lower value, upper value)
The point estimate is the middle point of the interval:
𝑙𝑜𝑤𝑒𝑟 𝑣𝑎𝑢𝑒 + 𝑢𝑝𝑝𝑒𝑟 𝑣𝑎𝑙𝑢𝑒
𝑝𝑜𝑖𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 =
2
The “margin” is half the width of the interval:
𝑢𝑝𝑝𝑒𝑟 𝑣𝑎𝑙𝑢𝑒 − 𝑙𝑜𝑤𝑒𝑟 𝑣𝑎𝑙𝑢𝑒
margin 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 =
2
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6. a. Suppose that a 95% confidence interval for p is (0.2638,0.5544). Find the point estimate for
this data and the “margin” of error for this confidence interval.
b. Published data general has a margin of error that is no more than 3%. In other words, we
need to estimate p to within 0.03. Using a 95% CI, how large a sample do we need to estimate the p
within 0.03?
7.
Cross-pollination can occur between farmers’ crop plants and wild populations, resulting in
hybrid plants. It is feared that when this happens the wild plant (that is, a weed) evolves into a
more sturdy plant and may have increased capability of survival and thus become more of a problem
for farmers. In a study to address this concern, botanists are planning to create a hybrid plant by
crossing common wheat plants with wild sunflowers.
a) The botanists would like to estimate p, the proportion of hybrid plants that survive, to
within 0.05 with 95% confidence. Records from previous research with wild non-hybrid sunflowers
indicate a survival proportion of 89%, and the botanists believe the survival proportion of the hybrid
plants will be close to this figure. If the 89% figure is accepted as a reasonable first estimate of the
survival proportion of hybrids, what sample size should they use for their study?
b) The botanists were fortunate in having past data to guide them with an initial estimate of
the proportion of hybrids that would survive to reproduce. In a few sentences, explain how your
procedure for choosing a sample size would differ from your method in part (a) if there were no
reasonable initial estimate of the survival proportion.
Homework: 10-13, 16-19, 21, 23, 25, 33
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9.3: Confidence Interval for a Population Mean
The general formula for a confidence interval for a population mean µ when:
1.
2.
3.
Cosmic radiation levels rise with increasing altitude, promoting researchers to consider how pilots
and flight crews might be affected by increased exposure to cosmic radiation. A study reported a
mean annual cosmic radiation dose of 219 mrems for a sample of flight personnel of Xinjiang
Airlines. Suppose this mean is based on a random sample of 100 flight crew members. Let s = 35
mrems.
Calculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation
exposure for Xinjiang flight crew members.
Conditions:
Calculation:
Conclusion:
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When 𝜎 is unknown, we use the sample standard deviation s to estimate 𝜎. In place of z-scores, we
must use:
Important Properties of t Distributions
t distributions are described by degrees of freedom (df)
1.
2.
3.
4.
The general formula for a confidence interval for a population mean µ based on a sample of size n
when:
1.
2.
3.
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The article “Chimps Aren’t Charitable” (Newsday, November 2, 2005) summarized the results of a
research study published in the journal Nature. In this study, chimpanzees learned to use an
apparatus that dispersed food when either of two ropes was pulled. When one of the ropes was
pulled, only the chimp controlling the apparatus received food. When the other rope was pulled,
food was dispensed both to the chimp controlling the apparatus and also a chimp in the adjoining
cage. The accompanying data represent the number of times out of 36 trials that each of seven
chimps chose the option that would provide food to both chimps (charitable response)
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24
19
20
20
Compute a 99% confidence interval for the mean number of charitable responses for the population
of all chimps.
Conditions:
Calculations:
Conclusion:
Choosing a Sample Size
The bound on error of estimation associated with a 95% confidence interval is
Solved for n:
Educated guess for 𝜎:
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The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a
particular university. For the estimate to be useful, it should be within $20 of the true population
mean. How large a sample should be used to be 95% confident of achieving this level of accuracy?
The financial aid office is believes that the amount spent on books varies with most values between
$150 to $550.
8. Data collected by Franz Boas in 1892 measure the heights of 1123 adult men from eight Native
American tribes of the Great Plains. These men had a mean height of 172.6 cm, with a standard
deviation of 6.2 cm. Assuming that the data is a random sample from the population, estimate the
mean height of all adult men from the Native American tribes of the Great Plains with a 95%
confidence interval:
A. Is the population type numerical or categorical?
B. How much confidence is desired? Write the formula for confidence interval.
C. Determine n, 𝑥̅ and s, and check that the sample meets the necessary assumptions.
D. Construct the CI.
E. Interpret your CI in the context of the problem.
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F. According to military records of the 19th century, Australian troops were tallest, with an average
height of 172 cm. Based on your CI results, does it appear that the adult men of the Great Plains
were the tallest in the world? Explain.
G. How large is the margin of error for your CI?
H. What sample size would be needed to estimate the mean height within 0.25 cm with 95&
confidence?
9. The concentration of mercury in lake waters is important to monitor. A recent accident at a
lakeside factor may have leaked mercury in the water. So a team of scientists measured the
mercury concentration (in mg/m3) at randomly selected locations in the lake. The data is given
below:
1.60
1.45 1.77 1.52 1.61 1.43 1.25 1.98 0.98 1.20
1.79
0.85 1.34 2.11 1.07
Estimate the mean mercury level in the lake with 90% confidence.
A. Construct the CI using the TInterval on your calculator:
TI 83/84: Press Stat, choose Tests, scroll down to TInterval
TI 89: In the Stats/List Editor, press F7 (Ints) and select T-Interval
B. Make a conclusion based on the CI
C. How does the mean mercury level in the lake compare to the maximum 2 mg/m3 limit set by the
EPA?
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10. The foraging capability of Downy Woodpeckers can be compromised if part of their habitat
disappears due to significant human encroachment. These creatures eat insects in trees, and if they
cannot find enough food due to a decreasing forest area their body mass may decrease. A previous
study of Downy Woodpeckers indicates that for male Downy Woodpeckers the standard deviation of
body mass is approximately 1.5g. Suppose that an estimate m , the current average mass of male
Downy Woodpeckers in a local forest preserve, is desired. If the previous study value is taken as a
reasonable first estimate of the population standard deviation, what sample size is necessary to
estimate m to within 0.2 grams with 95% confidence?
11. A simple random sample of the houses in a particular town was selected and the level of radon
gas is measured for each house in the sample. The values collected are given below in parts per
million (ppm). Experience has shown that radon gas level is approximately normally distributed in
this population.
4, 8, 13, 6, 7, 3, 9, 8, 11, 8
a) Construct and interpret a 99% confidence interval for m .
b) The city standards for radon call for a radon level of no more than 6 ppm. From these data, does
it appear that the mean radon level for city houses is less than 6 ppm? Provide statistical
justification for your response.
Homework: 34-37, 40-41, 43, 45, 48, 52
Extra Credit: 53-55
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