Transcript 10-1 Day 2

AP STATISTICS
LESSON 10 – 1
(DAY 2)
CONFIDENCE INTERVAL FOR
POPULATION MEAN
ESSENTIAL QUESTION:
What are the formulas used and
conditions necessary to construct
confidence intervals?
Objectives:
• To construct confidence intervals.
• To recognize conditions in which
confidence intervals can be used.
Confidence Interval for a
Population Mean
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 SAT Math score 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
Page 544
Finding z*
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.
Common Confidence Levels
Confidence levels
90%
95%
99%
tail area
0.05
0.025
0.005
z*
1.645
1.96
2.576
Notice 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 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.
Figure 10.6 Page 545
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 (1C)/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
Confidence Interval for a
Population Mean
Choose an SRS of size n from a population
having unknown mean μ and known
standard deviation of σ. A level C
confidence interval for μ is
x ± z*σ/√ n
Here z* is the value with C between –z*
and z* under the standard normal curve.
This interval is exact when the population
distribution is normal and is approximately
correct for large n in other cases.
Example 10.5 Page 546
Video Screen Tension
• 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