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Math 1680
Lesson #19: Chapter 23
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Chapter 23: Confidence Interval for Averages
Ex. #1: Twenty-five draws will be made at random with replacement
from the box with 7 tickets numbered “1,” “2,”  , “7.”
The sum of the draws will be approximately ____________, give or take
_______ or so.
Solution:
The average of the box is
The SD of the box is
1 2  3  4  5  6  7
4
7
(3) 2  (2) 2  (1) 2  0 2  12  2 2  32
2
7
So, the expected value of the sum is
The SE of the sum is
4  25  100
25  2  10
Ex. #2: The AVERAGE of the draws will be approximately
___________. Give or take ______ or so.
Solution:
Notice that the sum of the 25 draws is about 100, give or take 10 or so.
Thus, the average of the 25 draws is about 100  4 . (Not surprisingly,
25
this is the average of the box.)
The SE of the average is
10
 0 .4
25
Math 1680
Lesson #19: Chapter 23
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Formula: Expected value of average of draws = average of box
SE for average of draws =
SE for sum
number of draws
Just like the sum of draws, the normal curve can be used to approximate
the chance that the average lies in a specified range.
Ex. #3: Four hundred draws are made at random with replacement from
the box that contains seven tickets, numbered “1”, “2”,  , “7”. Find
the chance that the average of the draws will be more than 3.9.
Solution:
Expected value for average of draws = 4
SE for sum of draws =
400  2  40
SE for average of draws =
40
 0 .1
400
Chance  _________
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Lesson #19: Chapter 23
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Ex. #4: The cookie machine at Chips Ahoy adds a random number of
chips to each cookie. The number of chips is a random number with
average 28.5 and SD 5.3. Find the probability that, in a bag of 50
cookies, the average number of chips per cookie is at least 30.
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Math 1680
Lesson #19: Chapter 23
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Ex. #5: Two hundred draws are made at random with replacement from
the box that contains four tickets, 1 numbered “1”, two numbered “2”
and one numbered “3”. The average of the box is 2; while the SD is 1 .
2
True or False:
A) The expected value for the average of the draws is exactly 2.
B) The expected value for the average of the draws is around 2, give or
take 0.05 or so.
C) The average of the draws will be around 2, give or take 0.05 or so.
D) The average of the draws will be exactly 2.
E) The average of the box is exactly 2
F) The average of the box is around 2, give or take 0.05, or so.
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Lesson #19: Chapter 23
Page 5 of 9
SAMPLE AVERAGES
Ex. #6: A university has 25,000 registered students. In a survey of 318
students, the average age of the sample is found to be 22.3, with a
sample SD of 4.5 years. Find a 95% confidence interval for the average
age of all 25,000 students.
Question: What do we have to know about the sample before
proceeding?
Answer: Of course, we estimate the average of the population to be
22.3 years – but this estimate will not be exact. To determine the
magnitude of the error, we need to find the SE, and that means a box
model.
To find the SE of the sum of the draws, we use the bootstrap method
again: We estimate the SD of the box to be 4.5. So
SE for sum of draws
 318  4.5  80.2
SE for average of draws

years
SE for sum of draws
 0.25
318
years
So, the average age is 22.3, give or take 0.25 years or so.
Finally, the 95%--confidence interval is obtained by going 2 SEs either
way from the sample average:
22.3  2(0.25)  21.8 years
22.3  2(0.25)  22.8 years
In summary, the 95%--confidence interval is 21.8 years – 22.8 years.
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Math 1680
Lesson #19: Chapter 23
Page 6 of 9
Observations:
1) We are NOT saying that 95% of the students are between 21.8 and
22.8 years old – this is patently ridiculous, of course.
2) We are NOT saying that there is a 95% chance that the average age is
between 21.8 and 22.8 years. The population average is constant – it is
either in the range or it is not.
3) We ARE saying that if multiple people make such simple random
samples and obtain confidence intervals, then 95% of these confidence
intervals will contain the true average.
4) Remember: There is no such thing as a 100% confidence interval.
Ex. #7: A biological research team measures the weights of 54
chipmunks, randomly chosen. These chipmunks were found to have
average weight 8.7 ounces, with SD 1.4 ounces. Find a 68% confidence
interval for the average weight of chipmunks.
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Lesson #19: Chapter 23
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REVIEW
1) Probability deals with known box and unknown sample; whereas;
2) Statistics deals with unknown box and known sample.
Fundamental

SE for sum =
SE for average =
number of draws  SD of box 
SE for sum
number of draws
SE for count = SE for sum, from a
SE for percent =
0 1
box
SE count
 100%
number of draws
Reasoning from Box to Draws: SE is exactly computed from the
contents of the box.
Reasoning from Draws to Box: SD is estimated from the sample SD.
The population average is estimated from the sample average (likewise
with percentage).
The normal curve is used explicitly to find the chance that the sample
(sum, percentage, count or average) lies in a specified range, if there is a
large number of draws.
The normal curve is used implicitly to determine confidence intervals, if
the sample size is sufficiently large.
The logic above only applies for simple random samples.
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Lesson #19: Chapter 23
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Ex. #8: A box of tickets has an average of 80 and an SD of 10. Four
hundred draws will be made at random with replacement from this box.
Find the probability that the average of these draws is less than 81.
Ex. #9: Duracell tests 100 batteries in flashlights. They determine that
the average life of the batteries in this sample is 3.58 hours, with a
sample SD of 1.58 hours. Find a 95% confidence interval for the
average life of a Duracell battery in a flashlight.
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Math 1680
Lesson #19: Chapter 23
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Ex. #10: In a simple random sample of 750 households (in a city of
millions), the average number of TV sets in the sample households is
1.86, with an SD of 0.80. If possible, find a 95% confidence interval for
the average number of TV sets per household in the city.
Ex. #11: As part of a survey, all persons age 16 and over are
interviewed – a sample of 1,528 people. On average, these people
watched 5.2 hours of TV the Sunday before the survey, and the SD was
4.5 hours. If possible, find a 95%-confidence interval for the average
number of hours spent watching TV on that Sunday by all persons age
16 and over.
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