8-1 business statistics: a decision-making approach 8 th edition chapter 8 estimating single...
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8-1
Business Statistics: A Decision-Making Approach
8th Edition
Chapter 8Estimating Single
Population Parameters
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Chapter Goals
After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence
interval estimate Construct and interpret a confidence interval estimate for a
single population mean using both the z and t distributions Determine the required sample size to estimate a single
population mean or a proportion within a specified margin of error
Form and interpret a confidence interval estimate for a single population proportion
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Overview of the Chapter
Builds upon the material from Chapter 1 and 7 Introduces using sample statistics to estimate population
parameters Because gaining access to population parameters can be
expensive, time consuming and sometimes not feasible Confidence Intervals for the Population Mean, μ
when Population Standard Deviation σ is Known when Population Standard Deviation σ is Unknown
Confidence Intervals for the Population Proportion, p Determining the Required Sample Size for means and
proportions
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Estimation Process
(mean, μ, is unknown)
Population
Random Sample Mean
(point estimate)
Mean x = 50
Sample
I am 95% confident that μ is between 40 & 60.
confidenceinterval
Confidence Level
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Point Estimate
Suppose a poll indicate that 62% (sample mean) of the people favor limiting property taxes to 1% of the market value of the property.
The 62% is the point estimate of the true population of people who favor the property-tax limitation.
EPA tested average Automobile Mileage (point estimate)
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Confidence Interval
The point estimate (sample mean) is not likely to exactly equal the population parameter because of sampling error.
Probability of “sample mean = population mean” is zero Problem: how far the sample mean is from the
population mean. To overcome this problem, “confidence interval” can be
used as the most common procedure.
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Point and Interval Estimates
So, a confidence interval provides additional information about variability within a range.
The interval incorporates the sampling error
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of confidence interval
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Confidence Level
Confidence level = 95% Meaning: In the long run, 95% of all the confidence
intervals will contain the true parameter Describes how strongly we believe that a particular
sampling method will produce a confidence interval that includes the true population parameter.
A percentage (less than 100%) Most common: 90% (α = 0.1), 95% (α = 0.05), 99%
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Common Levels of Confidence
Commonly used confidence levels are 90%, 95%, and 99%
Confidence Level
Critical value, z
1.28
1.645
1.96
2.33
2.58
3.08
3.27
80%
90%
95%
98%
99%
99.8%
99.9%
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General Formula
The general formula for the confidence interval is:
Point Estimate Margin of Error
Margin of Error: Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval
(z or t - Value) * (Standard Error)
n
σzx
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Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall
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General Formula
According to our textbook
Point Estimate (Critical Value)(Standard Error)
z or t -value based on the level of
confidence desired
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Conditions for finding the Critical Value
The Central Limit Theorem states that the sampling distribution of a statistic will be normally (nearly normally) distributed if at least n > 30.
The sampling distribution of the mean is normally distributed when the population distribution is normally distributed.
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How to Find the z Value
When one of these conditions is satisfied, the critical value can be expressed as a z score.
To find the z value, see the next slide.
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Finding the z Value
Consider a 95% confidence interval:
n
σzx
-z = -1.96 z = 1.96
.0252
α .025
2
α
Point EstimateLower Confidence Limit
UpperConfidence Limit
z units:
x units: Point Estimate
0
1.96z
n
σzx x
0.95/2 (because of lower and upper limit) = 0.475
find on z table = 1.96
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When the population distribution type is unknown or when the sample size is small (less than 30), the t value is preferred.
The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after
sample mean has been calculated
d.f. = n – 1
Only n-1 independent pieces of data information left in the sample because the sample mean has already been obtained
How to Find the t value
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t Distribution:Try the t simulation one the website
t0
t (df = 5)
t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal
(t with df = )
Note: t compared to z as n increases
As n the estimate of s becomes better so t
converges to z
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t Table
Confidence Level
df 0.50 0.80 0.90
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920The body of the table contains t values, not probabilities
Let: n = 3 df = n - 1 = 2 confidence level: 90%
/2 = 0.05
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t Distribution Values
With comparison to the z value
Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____
0.80 1.372 1.325 1.310 1.28
0.90 1.812 1.725 1.697 1.64
0.95 2.228 2.086 2.042 1.96
0.99 3.169 2.845 2.750 2.58
Note: t compared to z as n increases
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Example
A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ
d.f. = n – 1 = 24, so
The confidence interval is
2.0639t0.025,241n,/2 t
25
8(2.0639)50
n
stx
46.698 53.302
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Summary of When to use z or t
Use z value (or score) Population is normally distributed (very rare and even
though sample size is small) Sample size is n > 30
Use t value (or score) Do not know population distribution type (not normal)
Do not know Pop. Mean and Std Many real world situations
Sample size is less than 30
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Using Analysis ToolPak
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Download “Confidence Interval Example” Excel file
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Using Analysis ToolPak
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Margin of Error
Confidence Interval
119.9 (mean) ± 2.59
117.31 ------ 122.49
small sample: apply “t” distribution automatically)
Don’t even worry about p* = 1 - α/2
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Using Analysis ToolPak(large sample: use normal (z) distribution automatically )
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Determining Required Sample Size
Wishful situation High confidence level, low margin of error, and right sample size
In reality, conflict among three…. For a given sample size, a high confidence level will tend to
generate a large margin of error For a given confidence level, a small sample size will result in an
increased margin of error Reducing of margin of error requires either reducing the confidence
level or increasing the sample size, or both
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Determining Required Sample Size
How large a sample size do I really need? Sampling budget constraint Cost of selecting each item in the sample
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Determining Required Sample SizeWhen σ is known
The required sample size can be found to reach a desired margin of error (e) and
level of confidence (1 - ) Required sample size, σ known:
2
222
e
σz
e
σzn
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Required Sample Size Example
If = 45 (known), what sample size is needed to be 90% confident of being correct within ± 5?
(Always round up)
219.195
(45)1.645
e
σzn
2
22
2
22
So the required sample size is n = 220
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If σ is unknown (most real situation)
If σ is unknown, three possible approaches
1. Use a value for σ that is expected to be at least as large as the true σ
2. Select a pilot sample (smaller than anticipated sample size) and then estimate σ with the pilot sample standard deviation, s
3. Use the range of the population to estimate the population’s Std Dev. As we know, µ ± 3σ contains virtually all of the data. Range = max – min.
Thus, R = (µ + 3σ) – (µ - 3σ) = 6σ. Therefore, σ = R/6 (or R/4 for a more conservative estimate, producing a larger sample size)
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Example when σ is unknown
Example 8-5: Jackson’s Convenience Stores Using a pilot sample approach Available on the class website
Make sure to review all the examples from page 306 to 327.
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Confidence Intervals for the Population Proportion, π
Try by yourself! An interval estimate for the population
proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ). For example, estimation of the proportion of
customers who are satisfied with the service provided by your company
Sample proportion, p = x/n
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Confidence Intervals for the Population Proportion, π
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:
(continued)
n
p)p(1sp
n
π)π(1σπ
See Chpt. 7!!
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Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size
n
p)p(1zp
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Example
A random sample of 100 people
shows that 25 are left-handed.
Form a 95% confidence interval for
the true proportion of left-handers
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Example
A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.
1.
2.
3.
0.0433 /1000.25(0.75)p)/np(1S
0.2525/100 p
p
0.3349 0.1651
(0.0433) 1.96 0.25
(continued)
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Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%
Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.
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Changing the sample size
Increases in the sample size reduce the width of the confidence interval.
Example: If the sample size in the above example is
doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at 0.25, but the width shrinks to
0.19 0.31
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Finding the Required Sample Size
for Proportion Problems
n
π)π(1ze
Solve for n:
Define the margin of error:
2
2
e
π)(1πzn
π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.50 – worst possible variation thus the largest sample size)
Will be in % units
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What sample size...?
How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)
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What sample size...?
Solution:For 95% confidence, use Z = 1.96e = 0.03p = 0.12, so use this to estimate π
So use n = 451
450.74(0.03)
.12)0(0.12)(1(1.96)
e
π)(1πzn
2
2
2
2
(continued)
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Using PHStat
PHStat can be used for confidence intervals for the mean or proportion
Two options for the mean: known and unknown population standard deviation
Required sample size can also be found Download from the textbook website
The link available on the class website
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PHStat Interval Options
options
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PHStat Sample Size Options
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Using PHStat (for μ, σ unknown)
A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ
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Using PHStat (sample size for proportion)
How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)
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Chapter Summary
Discussed point estimates Introduced interval estimates Discussed confidence interval estimation for
the mean [σ known] Discussed confidence interval estimation for
the mean [σ unknown] Addressed determining sample size for mean
and proportion problems Discussed confidence interval estimation for
the proportion
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