Transcript 9.1

Chapter 9
Section 1
The Logic in Constructing
Confidence Intervals about a
Population Mean where the
Population Standard Deviation is Known
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 1 of 39
Confidence Intervals
● Learning objectives
1

Compute a point estimate of the population mean
2 Construct and interpret a confidence interval about
the population mean (assuming the population
standard deviation is known)
3
 Understand the role of margin of error in constructing
a confidence interval
4
 Determine the sample size necessary for estimating
the population mean within a specified margin of error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 2 of 39
Confidence Intervals
● Learning objectives
1

Compute a point estimate of the population mean
2 Construct and interpret a confidence interval about
the population mean (assuming the population
standard deviation is known)
3
 Understand the role of margin of error in constructing
a confidence interval
4
 Determine the sample size necessary for estimating
the population mean within a specified margin of error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 3 of 39
Chapter 9 – Section 1
● The environment of our problem is that we want
to estimate the value of an unknown population
mean
● The process that we use is called estimation
● This is one of the most common goals of
statistics
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 4 of 39
Chapter 9 – Section 1
● Estimation involves two steps
 Step 1 – to obtain a specific numeric estimate, this is
called the point estimate
 Step 2 – to quantify the accuracy and precision of the
point estimate
● The first step is relatively easy
● The second step is why we need statistics
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 5 of 39
Chapter 9 – Section 1
● Some examples of point estimates are
 The sample mean to estimate the population mean
 The sample standard deviation to estimate the
population standard deviation
 The sample proportion to estimate the population
proportion
 The sample median to estimate the population
median
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 6 of 39
Confidence Intervals
● Learning objectives
1

Compute a point estimate of the population mean
2 Construct and interpret a confidence interval about
the population mean (assuming the population
standard deviation is known)
3
 Understand the role of margin of error in constructing
a confidence interval
4
 Determine the sample size necessary for estimating
the population mean within a specified margin of error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 7 of 39
Chapter 9 – Section 1
● The most obvious point estimate for the
population mean is the sample mean
● Now we will use the material in Chapter 8 on the
sample mean to quantify the accuracy and
precision of this point estimate
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 8 of 39
Chapter 9 – Section 1
● An example of what we want to quantify
 We want to estimate the miles per gallon for a certain
car
 We test some number of cars
 We calculate the sample mean … it is 27
 27 miles per gallon would be our best guess
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 9 of 39
Chapter 9 – Section 1
● How sure are we that the gas economy is 27
and not 28.1, or 25.2?
● We would like to make a statement such as
“We think that the mileage is 27 mpg
and we’re pretty sure that we’re
not too far off”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 10 of 39
Chapter 9 – Section 1
● A confidence interval for an unknown parameter
is an interval of numbers
 Compare this to a point estimate which is just one
number, not an interval of numbers
● The level of confidence represents the expected
proportion of intervals that will contain the
parameter if a large number of different samples
is obtained
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 11 of 39
Chapter 9 – Section 1
● What does the level of confidence represent?
● If we have a process for calculating confidence
intervals with a 90% level of confidence
 Assume that we know the population mean
 We then obtain a series of 50 random samples
 We apply our process to the data from each random
sample to obtain a confidence interval for each
● Then, we would expect that 90% of those 50
confidence intervals (or about 45) would contain
our population mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 12 of 39
Chapter 9 – Section 1
● If we expect that a method would create
intervals that contain the population mean 90%
of the time, we call those intervals
90% confidence intervals
● If we have a method for intervals that contain the
population mean 95% of the time, those are
95% confidence intervals
● And so forth
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 13 of 39
Chapter 9 – Section 1
● The level of confidence is always expressed as
a percent
● The level of confidence is described by a
parameter α
● The level of confidence is (1 – α) • 100%
 When α = .05, then (1 – α) = .95, and we have a 95%
level of confidence
 When α = .01, then (1 – α) = .99, and we have a 99%
level of confidence
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 14 of 39
Chapter 9 – Section 1
● To tie the definitions together (in English)
 We are using the sample mean to estimate the
population mean
 With each specific sample, we can construct a 95%
confidence interval
 As we take repeated samples, we expect that 95% of
these intervals would contain the population mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 15 of 39
Chapter 9 – Section 1
● To tie all the definitions together (using statistical
terms)
 We are using a point estimator to estimate the
population mean
 We wish to construct a confidence interval with
parameter α, the level of confidence is (1 – α) • 100%
 As we take repeated samples, we expect that
(1 – α) • 100% of the resulting intervals will contain
the population mean
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 16 of 39
Chapter 9 – Section 1
● Back to our 27 miles per gallon car
“We think that the mileage is 27 mpg
and we’re pretty sure that
we’re not too far off”
● Putting in numbers,
“We estimate the gas mileage is 27 mpg
and we are 90% confident that
the real mileage of this model of car
is between 25 and 29 miles per gallon”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 17 of 39
Chapter 9 – Section 1
“We estimate the gas mileage is 27 mpg”
● This is our point estimate
“and we are 90% confident that”
● Our confidence level is 90% (i.e. α = 0.10)
“the real mileage of this model of car”
● The population mean
“is between 25 and 29 miles per gallon”
● Our confidence interval is (25, 29)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 18 of 39
Chapter 9 – Section 1
● In this section, we assume that we know the
standard deviation of the population (σ)
● This is not very realistic … but we need it for
right now
● We’ll solve this problem in a better way (where
we don’t know what σ is) in the next section …
but first we’ll do this one
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 19 of 39
Chapter 9 – Section 1
● If n is large enough, i.e. n ≥ 30, we can assume
that the sample means have a normal
distribution with standard deviation σ / √ n
● We look up a standard normal calculation
 95% of the values in a standard normal are between
–1.96 and 1.96 … in other words between ± 1.96
● We now use this applied to a general normal
calculation
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 20 of 39
Chapter 9 – Section 1
● The values of a general normal random variable
are less than ± 1.96 times its standard deviation
away from its mean 95% of the time
● Thus the sample mean is within
± 1.96 σ / √ n
of the population mean 95% of the time
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 21 of 39
Chapter 9 – Section 1
● Because the sample mean has an
approximately normal distribution, it is in the
interval
  1.96 

n
around the (unknown) population mean 95% of
the time
● We can flip that around to solve for the
population mean μ
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 22 of 39
Chapter 9 – Section 1
● After we solve for the population mean μ, we
find that μ is within the interval
x  1.96 

n
around the (known) sample mean “95% of the
time”
● This isn’t exactly true in the mathematical sense
as the population mean is not a random variable
… that’s why we call this a “confidence” instead
of a “probability”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 23 of 39
Chapter 9 – Section 1
● Thus a 95% confidence interval for the mean is
x  1.96 

n
● This is in the form
Point estimate ± margin of error
● The margin of error here is 1.96 • σ / √ n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 24 of 39
Chapter 9 – Section 1
● For our car mileage example
 Assume that the sample mean was 27 mpg
 Assume that we tested 40 cars
 Assume that we knew that the population standard
deviation was 6 mpg
● Then our 95% confidence interval would be
27  1.96 
6
40
or 27 ± 1.9
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 25 of 39
Chapter 9 – Section 1
● If we wanted to compute a 90% confidence
interval, or a 99% confidence interval, etc., we
would just need to find the right standard normal
value
● Frequently used confidence levels, and their
critical values, are
 90% corresponds to 1.645
 95% corresponds to 1.960
 99% corresponds to 2.575
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 26 of 39
Chapter 9 – Section 1
● The numbers 1.645, 1.960, and 2.575 are
written as Z-values
 z0.05 = 1.645 … P(Z ≥ 1.645) = .05
 z0.025 = 1.960 … P(Z ≥ 1.960) = .025
 z0.005 = 2.575 … P(Z ≥ 2.575) = .005
where Z is a standard normal random variable
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 27 of 39
Chapter 9 – Section 1
● Why do we use α = 0.025 for 95% confidence?
● To be within something 95% of the time
 We can be too low 2.5% of the time
 We can be too high 2.5% of the time
● Thus the 5% confidence that we don’t have is
split as 2.5% being too high and 2.5% being too
low … needing α = 0.025 (or 2.5%)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 28 of 39
Chapter 9 – Section 1
● In general, for a (1 – α) • 100% confidence
interval, we need to find zα/2, the critical Z-value
● zα/2 is the value such that
P(Z ≥ zα/2) = α/2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 29 of 39
Chapter 9 – Section 1
● Once we know these critical values for the
normal distribution, then we can construct
confidence intervals for the sample mean
x  z / 2

n
to x  z / 2

n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 30 of 39
Chapter 9 – Section 1
● Learning objectives
1

Compute a point estimate of the population mean
2 Construct and interpret a confidence interval about
the population mean (assuming the population
standard deviation is known)
3
 Understand the role of margin of error in constructing
a confidence interval
4
 Determine the sample size necessary for estimating
the population mean within a specified margin of error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 31 of 39
Chapter 9 – Section 1
● If we write the confidence interval as
27 ± 2
then we would call the number 2 (after the ±) the
margin of error
● So we have three ways of writing confidence
intervals
 (25, 29)
 27 ± 2
 27 with a margin of error of 2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 32 of 39
Chapter 9 – Section 1
● The margin of error depends on three factors
 The level of confidence (α)
 The sample size (n)
 The standard deviation of the population (σ)
● We’ll now calculate the margin of error
● Once we know the margin of error, we can state
the confidence interval
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 33 of 39
Chapter 9 – Section 1
● The margin of errors would be
 1.645 • σ / √ n for 90% confidence intervals
 1.960 • σ / √ n for 95% confidence intervals
 2.575 • σ / √ n for 99% confidence intervals
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 34 of 39
Chapter 9 – Section 1
● Learning objectives
1

Compute a point estimate of the population mean
2 Construct and interpret a confidence interval about
the population mean (assuming the population
standard deviation is known)
3
 Understand the role of margin of error in constructing
a confidence interval
4
 Determine the sample size necessary for estimating
the population mean within a specified margin of error
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 35 of 39
Chapter 9 – Section 1
● Often we have the reverse problem where we
want an experiment to result in an answer with a
particular accuracy
● We have a target margin of error
● We need to find the sample size (n) needed to
achieve this goal
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 36 of 39
Chapter 9 – Section 1
● For our car miles per gallon, we had σ = 6
● If we wanted our margin of error to be 1 for a
95% confidence interval, then we would need
1.96 

6
 1.96 
1
n
n
● Solving for n would get us n = (1.96 • 6)2 or that
n = 138 cars would be needed
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 37 of 39
Chapter 9 – Section 1
● We can write this as a formula
● The sample size n needed to result in a margin
of error E for (1 – α) • 100% confidence is
 z / 2   
n

E


2
● Usually we don’t get an integer for n, so we
would need to take the next higher number (the
one lower wouldn’t be large enough)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 38 of 39
Summary: Chapter 9 – Section 1
● 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
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 9 Section 1 – Slide 39 of 39