Download Confidence Intervals

CHAPTER 10
Section 10.1 Part 2 – Estimating With Confidence
CONFIDENCE INTERVAL FOR A POPULATION MEAN
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When a sample of size n comes from a SRS, the
construction of the confidence interval depends on
the fact that the sampling distribution of the sample
mean 𝑥 is at least approximately normal.
This distribution is exactly normal if the population
is normal.
When the population is not normal, the central limit
theorem tells us that the sampling distribution of 𝑥
will be approximately normal if n is sufficiently
large.
CONDITIONS FOR CONSTRUCTING A CONFIDENCE
INTERVAL FOR 𝜇

The construction of a confidence interval for a
population mean 𝜇 is approximate when:

The data come from an SRS from the population of
interest, and

The sampling distribution of x is approximately
normal.
CONFIDENCE INTERVAL BUILDING STRATEGY
 Our construction of a 95% confidence interval
for the mean Mystery Mean began by noting
that any normal distribution has probability
about 0.95 within 2 standard deviations of its
mean.
 To
do that, we must go out z* standard
deviations on either side of the mean.
 Since
any normal distribution can be
standardized, we can get the value z* from the
standard normal table.
EXAMPLE 10.4 - FINDING Z*
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To find an 80% confidence interval, we must catch the
central 80% of the normal sampling distribution of 𝑥.
In catching the central 80% we leave out 20%, or 10% in
each tail.
So z* is the point with 10% area to its right.
z*
COMMON CONFIDENCE LEVELS
Confidence levels
90%
95%
99%
 Notice
tail area
0.05
0.025
0.005
z*
1.645
1.96
2.576
that for 95% confidence we use
z* = 1.960. This is more exact than the
approximate value z*= 2 given by the 68-9599.7 rule.
TABLE C


The bottom row of the C table (back cover of
book) can be used to find some values of z*.
Values of z* that mark off a specified area under
the standard normal curve are often called
critical values of the distribution.
EXAMPLE 10.6 CHANGING THE CONFIDENCE LEVEL


In general, the central probability C under a
standard normal curve lies between –z* and z*.
Because z* has area (1-C)/2 to its right under the
curve, we call it the upper (1-C)/2 critical value.
CRITICAL VALUE

The number z* with probability p lying to its right
under the standard normal curve is called the
upper p critical value of the standard normal
distribution.
LEVEL C CONFIDENCE INTERVALS
 Any
normal curve has probability C between
the points z* standard deviations below its
mean and the point z* standard deviations
above its mean.
 The
standard deviation of the sampling
distribution of 𝑥 is σ/√ n , and its mean is the
population mean 𝜇.

So there is probability C that the observed
sample mean 𝑥 takes a value between:
𝜇 – z*σ√ n and 𝜇 + σ/√ n
 Whenever
this happens, the population mean
𝜇 is contained between 𝑥 – z*σ√ n and 𝑥 +
z*σ/√n
EXAMPLE 10.5 - VIDEO SCREEN TENSION
 Read
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example 10.5 on p.546:
Step 1 – Identify the population of interest and the
parameter you want to draw conclusions about.
Step 2 – Choose the appropriate inference procedure.
Verify the conditions for using the selected procedure.
Step 3 – If conditions are met, carry out the inference
procedure.
Step 4 – Interpret your results in the context of the
problem.
Stem
Plot
Normal
Probability
Plot
INFERENCE TOOLBOX:
CONFIDENCE INTERVALS
To construct a confidence interval:

Step 1: Identify the population of interest and the
parameters you want to draw conclusions about.

Step 2: Choose the appropriate inference procedure.
Verify the conditions for the selected procedure.


Step 3: if the conditions are met, carry out the
inference procedure.
CI = estimate ± margin of error
Step 4: Interpret your results in the context of the
problem.
CONFIDENCE INTERVAL FORM
 The
form of confidence intervals for the
population mean 𝜇 rests on the fact that
the statistic 𝑥 used to estimate 𝜇 has a
normal distribution.
 Because
many sample statistics have
normal distributions (approximately),
confidence intervals have the form:
estimate ± z* σ estimate
HOW CONFIDENCE INTERVALS BEHAVE

Properties that are shared by all confidence
intervals:

The user chooses the confidence level and the margin
of error..

A small margin of error says that we have pinned
down the parameter quite precisely.
Margin of error = z*•σ/√n
MAKING THE MARGIN OF ERROR
SMALLER

z* and σ in the numerator and √n in the
denominator will make the margin of error
smaller when:

z* gets smaller. Smaller z* is the same as smaller
confidence level C.

There is a trade-off between the confidence level and
the margin of error.
MAKING MARGIN OF ERROR SMALLER
(CONTINUED…)
 To
obtain a smaller margin of error from
the same data you must be willing to
accept lower confidence:

σ gets smaller. The standard deviation σ measures
the variation in the population. You can think of the
variation among individuals in the population as noise
that obscures the average 𝜇. It is easier to pin down
𝜇 when σ is small.

n gets larger. Increasing the sample size n reduces
the margin of error for any fixed confidence level.
Because n appears under a square root sign, we must
take four times as many observations in order to cut
the margin of error in half.
EXAMPLE 10.6 - CHANGING THE CONFIDENCE LEVEL
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Suppose that the manufacturer in example 10.5 wants
a 99% confidence level.
Find the new margin of error, confidence level and
compare this level to the 90% confidence level found in
Example 10.5.

Read example 10.6 on p.550
CHOOSING THE SAMPLE SIZE
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A wise user of statistics never plans data
collection without planning the inference at the
same time.
You can arrange to have both high confidence
and a small margin of error by taking enough
observations.
EXAMPLE 10.7 DETERMINING SAMPLE SIZE
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Company management wants a report of the
mean screen tension for the day’s production
accurate to within ± 5 mV with a 95% confidence.
How large a sample of video monitors must be
measured to comply with this request?
Read example 10.7 on p.551-552
SAMPLE SIZE FOR
DESIRED MARGIN OF ERROR

To determine the sample size n that will yield a
confidence interval for a population mean with a
specified margin of error m, set the expression for the
margin of error (m) to be less than or equal to m and
solve for n:
z*•σ/√n ≤ m

In practice, taking observations costs time and money.
Do notice once again that it is the size of the sample
that determines the margin of error. The size of the
population does not determine the size of the sample
we need.
SOME CAUTIONS
 Any
formula for inference is correct only in specific
circumstances.
 The
data must be an SRS from the population.
 The
formula is not correct for probability sampling
designs more complex than an SRS.
 There
is no correct method for inference from data
haphazardly collected with bias of unknown size.
 Because
𝑥 is strongly influenced by a few extreme
observations, outliers can have a large effect
on the confidence interval. You should search
for outliers and try to correct them or justify their
removal before computing the interval.
CAUTIONS (CONTINUED…)
 If
the sample size is small and the population
is not normal, the true confidence level will be
different from the value C used in computing
the interval.

Examine your data carefully for skewness and
other signs of nonnormality. When n ≥15, the
confidence level is not greatly disturbed by
nonnormal populations unless extreme outliers or
quite strong skewness are present.
CAUTIONS (CONTINUED…)
 You
must know the standard deviation σ
of the population. This unrealistic
requirement renders the interval 𝑥 ±σ/√n
of little use in statistical practice.
 The margin of error in a confidence
interval covers only random sampling
errors.
 Practical difficulties, such as
undercoverage and nonresponse in a
sample survey, can cause additional
errors that may be larger than random
sampling error.

Homework: p.542- #’s 4, 6, 10, 13, 18, 22 & 24