Unit%206:%20Confidence%20Intervals

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Unit 6 Confidence Intervals
Elementary Statistics Larson Farber
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Definition Review
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Big Picture Confidence Intervals

• A group of college students collected data on the
  speed of vehicles traveling through a
  construction zone on a state highway, where the
  posted speed was 25 mph. Assume that the
  standard deviation for the recorded speed of the
  vehicles is 3.5 mph. The recorded speed of 14
  randomly selected vehicles is as follows 20,
  24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40,
  40
• Assuming speeds are approximately normally
  distributed, how fast do you think the true mean
  speed of drivers in this construction zone is?
•  

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Construction of a Confidence Interval

• The construction of a confidence interval for the
  population mean depends upon three factors
• The point estimate of the population
• The level of confidence
• The standard deviation of the sample mean

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Section 6.1
Confidence Intervals for the Mean (large samples)
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Point Estimate
DEFINITION A point estimate is a single value
estimate for a population parameter. The best
point estimate of the population mean is the
sample mean
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Part I Point Estimate
A random sample of 35 airfare prices (in dollars)
for a one-way ticket from Atlanta to Chicago.
Find a point estimate for the population mean, ??.
99 101 107
102 109 98
105 103 101
105 98 107
104 96 105
95 98 94
100 104 111
114 87 104
108 101 87
103 106 117
94 103 101
105 90
The sample mean is
The point estimate for the price of all one way
tickets from Atlanta to Chicago is 101.77.
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Interval Estimates
Point estimate
An interval estimate is an interval or range of
values used to estimate a population parameter.
The level of confidence, x, is the probability
that the interval estimate contains the
population parameter.
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Distribution of Sample Means
When the sample size is at least 30, the sampling
distribution for is normal.
Sampling distribution of
For c 0.95
0.95
0.025
0.025
z
0
-1.96
1.96
95 of all sample means will have standard scores
between z -1.96 and z 1.96
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Solution Finding the Margin of Error
?zc
zc 1.96
-zc -1.96
95 of the area under the standard normal curve
falls within 1.96 standard deviations of the
mean.
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Maximum Error of Estimate
The maximum error of estimate E is the greatest
possible distance between the point estimate and
the value of the parameter it is estimating for a
given level of confidence, c. When n is
greater than 30, the sample standard deviation,
s, can be used for .
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Part 2 Maximum Error of Estimate
A random sample of 35 airfare prices (in dollars)
for a one-way ticket from Atlanta to Chicago.
99 101 107
102 109 98
105 103 101
105 98 107
104 96 105
95 98 94
100 104 111
114 87 104
108 101 87
103 106 117
94 103 101
105 90
Find E, the maximum error of estimate for the
one-way plane fare from Atlanta to Chicago for a
95 level of confidence given s 6.69.
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Maximum Error of Estimate
Find E, the maximum error of estimate for the
one-way plane fare from Atlanta to Chicago for a
95 level of confidence given s 6.69.
s 6.69 n 35
Using zc 1.96,
You are 95 confident that the maximum error of
estimate is 2.22.
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Confidence Intervals for the Population Mean

• A c-confidence interval for the population mean µ
  
• The probability that the confidence interval
  contains µ is c.

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Part IIIConfidence Intervals for
Find the 95 confidence interval for the one-way
plane fare from Atlanta to Chicago.
You found 101.77 and E 2.22
Right endpoint
Left endpoint
103.99
99.55
With 95 confidence, you can say the mean one-way
fare from Atlanta to Chicago is between 99.55
and 103.99.
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How could we get closer?
103.99
99.55
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Construction of a Confidence Interval

• The construction of a confidence interval for the
  population mean depends upon three factors
• The point estimate of the population
• The level of confidence
• The standard deviation of the sample mean

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How could we get closer?
103.99
99.55

• Two ways to get a smaller Confidence Interval
• Lower confidence level (e.g. 75)
• Bigger Sample

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Sample Size
Given a c-confidence level and an maximum error
of estimate, E, the minimum sample size n, needed
to estimate , the population mean is
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Part IV Sample Size
You want to estimate the mean one-way fare from
Atlanta to Chicago. How many fares must be
included in your sample if you want to be 95
confident that the sample mean is within 2 of
the population mean?
You should include at least 43 fares in your
sample. Since you already have 35, you need 8
more.
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Section 6.2
What happens if we donthave 30 observations?
Confidence Intervals for the Mean (small samples)
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Normal or t-Distribution?
Is n ? 30?
Is the population normally, or approximately
normally, distributed?
Cannot use the normal distribution or the
t-distribution.
Is ? known?
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• Comparing three curves
• The standard normal curve
• The t curve with 14 degrees of freedom
• The t curve with 4 degrees of freedom

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The t-Distribution
If the distribution of a random variable x is
normal and n lt 30, then the sampling distribution
of is a t-distribution with n 1 degrees
of freedom.
Sampling distribution
.90
.05
.05
t
0
The critical value for t is 1.782. 90 of the
sample means (n 13) will lie between t
-1.782 and t 1.782.
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Confidence IntervalSmall Sample
Maximum error of estimate
In a random sample of 13 American adults, the
mean waste recycled per person per day was 4.3
pounds and the standard deviation was 0.3 pound.
Assume the variable is normally distributed and
construct a 90 confidence interval for .
1. The point estimate is 4.3 pounds
2. The maximum error of estimate is
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Finding tc

• If c 0.90
• n 13 (df 12)
• tc ?
• d.f. n - 1

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http//surfstat.anu.edu.au/surfstat-home/tables/t.
php
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Confidence IntervalSmall Sample
Maximum error of estimate
In a random sample of 13 American adults, the
mean waste recycled per person per day was 4.3
pounds and the standard deviation was 0.3 pound.
Assume the variable is normally distributed and
construct a 90 confidence interval for .
1. The point estimate is 4.3 pounds
2. The maximum error of estimate is
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Confidence IntervalSmall Sample
1. The point estimate is 4.3 pounds
2. The maximum error of estimate is
Right endpoint
Left endpoint
)
(
4.152
4.448
4.15 lt lt 4.45
With 90 confidence, you can say the mean waste
recycled per person per day is between 4.15 and
4.45 pounds.
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Normal or t-Distribution?
Is n ? 30?
Is the population normally, or approximately
normally, distributed?
Cannot use the normal distribution or the
t-distribution.
Is ? known?
See Pg 329 of textbook
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1. The Graduate Management Admission Test (GMAT) is
   a test required for admission into many masters
   of business administration (MBA) programs. Total
   scores on the GMAT are normally distributed and
   historically have a standard deviation of 113.
   Suppose a random sample of 8 students took the
   test, and their scores are recorded.
2. Sean is estimating the average number of
   Christmas Trees he will find in the windows of
   each store in the mall. He observes each of the
   10 stores in the mall and records a sample mean
   of 15 trees with a standard deviation of 6.
3. Patrick wonders about the average number of
   servings of eggnog at the Holiday Party. He
   knows that typically this variable has a standard
   deviation of 2.2 servings. He records a sample
   mean of 4 servings for a sample of 50 people.

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1. The Graduate Management Admission Test (GMAT) is
   a test required for admission into many masters
   of business administration (MBA) programs. Total
   scores on the GMAT are normally distributed and
   historically have a standard deviation of 113.
   Suppose a random sample of 8 students took the
   test, and their scores are recorded. (We know
   population is normally distributed, so we can use
   Z even though n lt30)
2. Sean is estimating the average number of
   Christmas Trees he will find in the windows of
   each store in the mall. He observes each of the
   10 stores in the mall and records a sample mean
   of 15 trees with a standard deviation of 6. (We
   do not know population is normally distributed,
   so must use t with 9 degrees of freedom because
   we have a small sample and we do not know sigma)
   
3. Patrick wonders about the average number of
   servings of eggnog at the Holiday Party. He
   knows that typically this variable has a standard
   deviation of 2.2 servings. He records a sample
   mean of 4 servings for a sample of 50 people. (We
   do not know population is normally distributed,
   but we know sigma is 2.2 so we can use Z.)

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Section 6.3
Confidence Intervals for Population Proportions
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What if we are interested in a population
proportion or percentage?For example What
percentage of the population likes spinach?
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Confidence Intervals forPopulation Proportions
The point estimate for p, the population
proportion of successes, is given by the
proportion of successes in a sample
(Read as p-hat)
is the point estimate for the proportion of
failures where
Required Condition If np gt 5 and nq gt5
the sampling distribution for p-hat is normal.
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Confidence Intervals for Population Proportions
The maximum error of estimate, E, for a
x-confidence interval is
A c-confidence interval for the population
proportion, p, is
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Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449
were alcohol related. Construct a 99 confidence
interval for the proportion of fatal traffic
accidents that are alcohol related.
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Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449
were alcohol related. Construct a 99 confidence
interval for the proportion of fatal traffic
accidents that are alcohol related.
1. The point estimate for p is
2. 1907(.235) gt 5 and 1907(.765) gt 5, so the
sampling distribution is normal.
3.
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Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449
were alcohol related. Construct a 99 confidence
interval for the proportion of fatal traffic
accidents that are alcohol related.
Left endpoint
Right endpoint
)
(
.21
.26
0.21 lt p lt 0.26
With 99 confidence, you can say the proportion
of fatal accidents that are alcohol related is
between 21 and 26.
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Minimum Sample Size
If you have a preliminary estimate for p and q,
the minimum sample size given a x-confidence
interval and a maximum error of estimate needed
to estimate p is
If you do not have a preliminary estimate, use
0.5 for both .
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ExampleMinimum Sample Size
You wish to estimate the proportion of fatal
accidents that are alcohol related at a 99 level
of confidence. Find the minimum sample size
needed to be be accurate to within 2 of the
population proportion.
With no preliminary estimate use 0.5 for
You will need at least 4415 for your sample.
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ExampleMinimum Sample Size
You wish to estimate the proportion of fatal
accidents that are alcohol related at a 99 level
of confidence. Find the minimum sample size
needed to be be accurate to within 2 of the
population proportion. Use a preliminary estimate
of p 0.235.
With a preliminary sample you need at least n
2981 for your sample.
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Example 1 (pg 310)

• Market researchers use the number of sentences
  per advertisement as a measure of readability for
  magazine advertisements. Suppose for the 50
  advertisements we determine that the average
  number of sentences (xbar) is 12.4 and the
  standard deviation is 5.0
• Compute the 95 confidence interval for the mean
  mu.
• Question 1 Do we know the population standard
  deviation?
• Question 2 What is the interval?

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Solution

• xbar 12.4
• s 5.0
• n 50
• c 0.95 Zc 1.96 (for c 0.95)
• E 1.96 5 / v50 1.4
• 12.4 1.4 lt µ lt 12.4 1.4
• 11.0 lt µ lt 13.8
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• Answer We are 95 confident that the mean
  number of sentences in the POPULATION is between
  11.0 and 13.8.

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Example 2 (Pg. 327)

• You randomly select 16 coffee shops and measure
  the temperature of the coffee sold at each. The
  sample mean temperature is 162 degrees with a
  standard deviation of 10 degrees. You know that
  the distribution of temperature is normally
  distributed.
• Find the 95 confidence interval for the mean
  temperature.
• Find the 99 confidence interval for the mean
  temperature.

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95 Confidence Interval

• xbar 162.0
• s 10.0
• n 16
• c 0.95 tc 2.132 (df 15, c .95)
• E 2.132 10 / v16 5.3
• 162 5.3 lt µ lt 162 5.3
• 156.7 lt µ lt 167.3
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• Answer We are 95 confident that the average
  temperature of all the coffee in the POPULATION
  is between 157 and 167 degrees. 

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What about 99?

• xbar 162.0
• s 10.0
• n 16
• c 0.99 tc 2.947 (df 15, c .99)
• E 2.947 10 / v16 7.4
• 162 7.4 lt µ lt 162 7.4
• 154.6 lt µ lt 169.4
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• Answer We are 95 confident that the average
  temperature of all the coffee in the POPULATION
  is between 154.6 and 169.4 degrees.