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How much sleep did you get last night?
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29%
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14%
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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Sleep and Technology


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“43 percent of Americans say they rarely or never
get a good night’s sleep during the week”
“Nearly everyone, 95 percent, use electronics
(like TV, computer, or cell phone) within the hour
just before bed”
“Researchers caution that the use of such
devices are particularly harmful to the sleeponset process, since the artificial light can
suppress the release of melatonin which is our
sleep hormone.”
http://www.scientificamerican.com/podcast/episod
e.cfm?id=electornic-gadgets-before-bed-can-hCopyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 2
Chapter 23
Inference About Means
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
A Confidence Interval for Means? (cont.)
One-sample t-interval for the mean

When the conditions are met, we are ready to find the
confidence interval for the population mean, μ.

The confidence interval is

n 1
where the standard error of the mean is
y t
 SE  y 
s
SE  y  
n

The critical value tn*1 depends on the particular confidence
level, C, that you specify and on the number of degrees of
freedom, n – 1, which we get from the sample size.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 4
What About Spread? The Standard Deviation
(cont.)

The standard deviation, s, is just the square root
of the variance and is measured in the same
units as the original data.
 y  y 
2
s
n 1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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HW-11 Problem 5

A nutrition lab test 50 ‘reduced sodium’ hot dogs,
finding that the mean sodium content is 318mg
with a standard deviation of 36mg.

You want to create a 95% confidence interval to
test your hypothesis.

What assumptions have you made? Are these
assumptions correct?
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 6
What assumption have we made in this
inference?
1.
2.
3.
We have assumed that the hot dogs weights are
NOT multimodal and that the distribution of the
population of hot dog weights does not contain
any outliers.
We have assumed that the hot dog weights are
random and that the distribution of the
population of hot dog weights is not biased.
We have assumed that the hot dog weights are
independent and that the distribution of the
population of hot dogs is normal
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 7
“A lab tests 50 hot dogs”. Is the
randomization condition satisfied?
1.
2.
3.
4.
No, because the hot dogs came from the same
package
No, there is evidence to believe that the hot
dogs were not sampled at random
Yes, there is definitely evidence to believe that
the hot dogs were sampled at random
We don’t know that the hot dogs were sampled
at random, but it is reasonable to think that they
were
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 8
Is the 10% condition satisfied?
1.
2.
3.
4.
No, 50 hot dogs are more than 10% of all hot
dogs.
Yes, 50 hot dogs are more than 10% of all hot
dogs.
No, 50 hot dogs are less than 10% of all hot
dogs.
Yes, 50 hot dogs are less than 10% of all hot
dogs.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 9
Explain clearly what your interval
means
1.
2.
3.
4.
95% of the sodium content in this type of
‘reduced sodium’ hot dog will be contained in the
interval
We are 95% confident that the interval contains
the true mean sodium content for this type of hot
dog.
The interval contains the true mean sodium
content in this type of ‘reduce sodium’ hot dogs
95% of the time.
95% of all ‘reduced sodium’ hot dogs will have a
mean sodium content that falls within the interval
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 10
HW 11 - Problem 7

Students investigating the packaging potato chips
purchased 6 bags of chips marked with a net
weight of 28.1 grams.

They weighed the contents of each bag,
recording the weight as follows:

29.2, 28.2, 29.1, 28.5, 28.8, 28.6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 11
%
Is the randomization condition
satisfied?
1.
2.
3.
We don’t know that the bag of chips were
sampled at random, but it is reasonable to think
that they were
No, the 6 bags were not selected at random, but
it is reasonable to think that these bags are
representative of the population
Yes, there is definitely evidence to believe that
the bags of chips were sampled at random.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 12
%
Is the 10% condition satisfied?
1.
2.
3.
4.
Yes, 6 bags of chips are more than 10% of the
populations of all bags of chips
Yes, 6 bags of chips are less than 10% of the
populations of all bags of chips
No, 6 bags of chips are more than 10% of the
populations of all bags of chips
No, 6 bags of chips are less than 10% of the
populations of all bags of chips
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 13
Data: 29.2, 28.2, 29.1, 28.5, 28.8, 28.6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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%
This data is on the weight of a bag of
potato chips. Interpret the 95% CI
1.
2.
3.
4.
95% of all bags of chips will have a mean weight
that falls within the interval
95% of the chips will be contained in the interval
The interval contains the true mean weight of
the contents of a bag of chips 95% of the time
We are 95% confident that the interval contains
the true mean weight of the contents of a bag of
chips.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 15
Comment on the company’s state net
weight of 28.1g
1.
2.
3.
Since the interval is ABOVE the stated weight of
28.1 grams, there is evidence that the company
is filling the bags to MORE than the stated
weight, ON AVERGAGE.
Since the interval is BELOW the stated weight of
28.1 grams, there is evidence that the company
is filling the bags to LESS than the stated
weight, ON AVERGAGE.
Since the interval CONTAINS the stated weight
of 28.1 grams, there is evidence that the
company is filling the bags to the stated weight,
ON AVERGAGE.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 16
HW11- Problem 10
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Slide 1- 17
Students purchased random bags of
cookies from different stores. Is the
randomization condition met?
100%
1.
2.
Yes
No
0%
1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
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Students purchased cookies from
different stores. Is the independence
assumption met?
1.
2.
Yes
No
91%
9%
1
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2
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Is the 10% condition met?
1.
2.
Yes
No
100%
0%
1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
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Is the data nearly normal?
1.
2.
Yes
No
100%
0%
1
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 21
Data – Create a 95% CI
1022
1142
1120
1269
1276
1228
1202
1317
1325
1491
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The company claims at least 1000
Chips in EVERY bag. What would you
conclude?
69%
1.
2.
3.
The company’s claim
is true
The company’s claim
is false
We cannot test this
claim
31%
0%
1
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3
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HW 11 - Problem 9
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One of the important factors in auto safety is the
weight of the vehicle.
Insurance companies are interested in knowing
the average weight of cars currently licensed.
They believe it is 3,000 lbs. (i.e. hypothesize).
To test this belief, they checked a random sample
of 91 cars and found:
Mean weight 2,855lbs.
SD 531.5lbs
Is this strong evidence that the mean weight of all
cars is NOT 3,000lbs.?
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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%
Is this strong evidence that the mean
weight of all cars is not 3000lbs?
1.
2.
3.
4.
Yes, there is sufficient evidence the mean is
different from 3000
No, there is sufficient evidence the mean is
different from 3000
Yes, there is NOT sufficient evidence the mean
is different from 3000
No, there is NOT sufficient evidence the mean is
different from 3000
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 25
HW 11 -Problem 11
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 26
Test the hypothesis that the mean
completion time for this maze is 60 sec
1.
2.
3.
4.
Reject null, there is sufficient evidence to
suggest the mean time is NOT 60 sec.
Reject null, there is NOT sufficient evidence to
suggest the mean time is NOT 60 sec.
Fail to reject null, there is sufficient evidence to
suggest the mean time is NOT 60 sec.
Fail to reject null, there is NOT sufficient
evidence to suggest the mean time is NOT 60
sec.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Eliminate the outlier then, test the
hypothesis that the mean completion
time for this maze is 60 sec
1.
2.
3.
4.
Reject null, there is sufficient evidence to
suggest the mean time is NOT 60 sec.
Reject null, there is NOT sufficient evidence to
suggest the mean time is NOT 60 sec.
Fail to reject null, there is sufficient evidence to
suggest the mean time is NOT 60 sec.
Fail to reject null, there is NOT sufficient
evidence to suggest the mean time is NOT 60
sec.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Do you think THIS maze meets the
“one-minute average” requirement?
There is NOT evidence that the mean time required
0% for rats to complete the maze is different from 60s.
The maze meets the requirements.
2.
There is evidence that the mean time required for
10%rats to complete the maze is different from 60s. The
maze meets the requirements.
3.
There is evidence that the mean time required for
90%
rats to complete the maze is different from 60s. The
maze DOES NOT meet the requirements.
4.
0% There is NOT evidence that the mean time required
for rats to complete the maze is different from 60s.
The maze DOES NOT meet the requirements.
1.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 29