Download Confidence Interval

CONFIDENCE INTERVALS of Means
AP STATISTICS, CHAPTER 19
Mrs. Watkins
INFERENCE:
Allows us to estimate population
parameters based on results of
one sample
We “infer” that something is true
about the population.
Purpose of a Confidence
Interval
Using a sample to estimate
the true value of a
population parameter, like
the mean μ.
Estimators:
A. Point estimates: X as an estimate of μ
B. Interval estimates: a range of values over
which the true parameter will be located
EX: 165.7 + 18.2 __________________
4.25 + 1.5 ___________________
Confidence Interval
(ESTIMATE) + MARGIN OF
ERROR
A 95% confidence interval is
7.5 + 0.85
We are 95% confident that the true
population mean lies between
_________ and ________.
If we wanted to be more confident, then
the interval will be _______________.
“the net”
Wider intervals give more CONFIDENCE.
Narrower intervals give more PRECISION.
Confidence Levels:
90% means 10% left out, so 5% on each side.
Invnorm(.05) = -1.65, so this is
what we call the Critical Value
95% --what would critical value be?
99% --what would critical value be?
FORMULA for Z Confidence Interval
for Means



*
X
X z 

 n
EXAMPLE #1
Estimate the mean size of a chocolate chip
cookie using a 95% confidence interval.
1. Formula—what numbers do we need?
2. Meaning—what does the final answer
indicate?
Meaning of the Confidence Interval:
We are 95% confident that the
true population parameter
(mean or proportion) is
between _____________ and
_____________.
EXAMPLE #2
A psychologist is doing a study on teen and sleep
patterns. Prior studies show that teens should
get 8 hours of sleep per night. A recent
sample of 15 students indicates that the mean
hours of sleep is 6.8 with a standard deviation
of 1.25 hours. Calculate a 90% confidence
intervals for the mean hours of sleep. Then do
a 99% confidence interval.
Important questions about
Confidence Intervals
1. How sure are we of that result?
2. What is our margin of error?
3. What effect does sample size have
on our confidence in our prediction?
Meaning of the Confidence Level:
In all possible samples of size n,
the true population mean will
be contained in 95% of the
intervals generated.
Assumptions for Z conf. int.
1. Random Sample—usually stated
2. Population Standard Deviation
KNOWN
3. Approximately normally distributed
**May be stated or you may have to
check box plot, normal prob plot or
mean and median
t confidence intervals
We use a t distribution when we do NOT
know the population standard deviation.
We RARELY know the population standard
deviation, so most confidence intervals
for the MEAN will be t intervals.
What is difference between z and t?
Critical Values are different because t is
actually a series of symmetric, mound
shaped curves.
Values depend on DEGREES of FREEDOM,
df = n – 1
where n is sample size.
T distribution
Formula for t confidence interval
s


*
x
X t 

 n
Different steps? Not much.
1. Different critical value
2. Same interpretation
3. Assumptions for t interval—very similar
a. Random Sample? Usually stated
b. Approximately normal? Usually have to check
c. Population st. deviation unknown
Find the critical values
1. Z int, 95%, n = 12
2. t int, 95%, n = 12
3. t int, 98%, n = 22
4. Z int, 92%, n = 15
5. t int, 90%, n = 62
6. Z int, 94%, n = 30