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Lesson 9 - 1
Logic in Constructing Confidence
Intervals about a Population Mean
where the Population Standard
Deviation is Known
Objectives
• Compute a point estimate of the population mean
• Construct and interpret a confidence interval about
the population mean (assuming population σ is
known)
• Understand the role of margin of error in
constructing a confidence interval
• Determine the sample size necessary for estimating
the population mean within a specified margin of
error
Vocabulary
• Point Estimate – value of a statistic that estimates the value of a
population parameter
• Confidence Interval – for an unknown parameter is an interval of
numbers (that the unknown falls between)
• Level of Confidence – represents the expected proportion of
intervals that will contain the parameter if a large number of
samples is obtained. The level of confidence is denoted by
(1- α) * 100%
• α – represents the percentage the parameter falls outside the
confidence interval (later known as a Type I error)
• Robust – minor departures from normality do not seriously affect
results
• Z-interval – confidence interval
Confidence Interval Estimates
Point estimate (PE) ± margin of error (MOE)
Point Estimate
Sample Mean for Population Mean
Sample Proportion for Population Proportion
Expressed numerically as an interval [LB, UB]
where LB = PE – MOE and UB = PE + MOE
Graphically:
PE
MOE
MOE
_
x
Margin of Error Factors
• Level of confidence: as the level of confidence
increases the margin of error also increases
• Sample size: as the sample size increases the
margin of error decreases (from Law of Large
Numbers)
• Population Standard Deviation: the more spread the
population data, the wider the margin of error
• MOE is in the form of
measure of confidence • standard dev / sample size
PE
MOE
MOE
_
x
Margin of Error, E
The margin of error, E, in a (1 – α) * 100% confidence
interval in which σ is known is given by
E = zα/2
σ
--n
where n is the sample size and zα/2 is the critical z-value.
Note: The sample size must be large (n ≥ 30) or the
population must be normally distributed.
Using Standard Normal
Measure of Confidence
Z critical value:
A value of the Z-statistic that corresponds to α/2
(1/2 because of two tails) for an α level of confidence
Level of Confidence
(1-α)
Area in each Tail
(α/2)
Critical Value
Z α/2
90%
0.05
1.645
95%
0.025
1.96
99%
0.005
2.575
PE
MOE
MOE
_
x
Interpretation of a Confidence Interval
A (1-α) * 100% confidence interval indicates that , if we
obtained many simple random samples of size n
from the population whose mean, μ, is unknown,
then approximately (1-α) * 100% of the intervals will
contain μ.
Note that is not a probability, but a level of the
statistician’s confidence.
Assumptions for Using Z CI
• Sample: simple random sample
• Sample Population: sample size must be large
(n ≥ 30) or the population must be normally
distributed.
Dot plots, histograms, normality plots and box
plots of sample data can be used as evidence if
population is not given as normal
• Population σ: known (If this is not true on AP
test you must use t-distribution!)
A (1 – α) * 100% Confidence Interval
about μ, σ Known
Suppose a simple random sample of size n is taken from
a population with an unknown mean μ and known
standard deviation σ. A (1 – α) * 100% confidence
interval for μ is given by
Lower bound = x – zα/2
σ
--n
where zα/2 is the critical z-value.
Upper bound = x + zα/2
σ
--n
Example 1
We have test 40 new hybrid SUVs that GM is resting its
future on. GM told us the standard deviation was 6 and
we found that they averaged 27 mpg highway. What
would a 95% confidence interval about average miles
per gallon be?
PE ± MOE
X-bar ± Z 1-α/2 σ / √n
27 ± (1.96) (6) / √40
LB = 25.141 < μ < 28.859 = UB
95% confident that the true average mpg (μ) lies between LB and UB
Example 2
GM told us the standard deviation for their new hybrid
SUV was 6 and we wanted our margin of error in
estimating its average mpg highway to be within 1
mpg. How big would our sample size need to be?
(Z 1-α/2 σ)²
n = ------------MOE²
MOE = 1
n = (Z 1-α/2 σ )²
n = (1.96∙ 6 )² = 138.3
n = 139
Summary and Homework
• Summary
– We can construct a confidence interval around a
point estimator if we know the population standard
deviation σ
– The margin of error is calculated using σ, the sample
size n, and the appropriate Z-value
– We can also calculate the sample size needed to
obtain a target margin of error
• Homework
– pg 458 – 465; 1-3, 10, 11, 19, 21, 23, 26, 37, 40, 47
Homework
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1: sample size, confidence level, and standard deviation
2: we widen our interval to become more confident that the true value
is in there
3: the greater the sample size the more the law of large numbers
helps assure the point estimate is closer to the population parameter
10: No, it does not look normal -- normality plot questionable
11: Yes, normality plot ok and box plot symmetric-like
19: σ= 13, x-bar=108, n=25
21
23
26: PE=103.4 minutes, c)(98.2,108.6) d) (99.0,107.8) e) decreased
37:
40: n=670
47: 4 times