# Confidence Intervals ```AP Statistics
Friday, 29 January 2016
• OBJECTIVE TSW determine confidence intervals.
• Yesterday’s tests are not graded.
• TEST: Continuous Distributions tests are graded.
• Up to 5 points will be awarded on the Continuous
Distributions test.
• Directions must be followed.
–
–
–
–
5 decimals for probabilities
3 decimals for other values
All work – including calculator key strokes – shown.
• This is due on Tuesday, 02 February 2016.
AP Exam Registration
• Registration must be done on line.
www.TotalRegistration.net/AP/443381
• Regular registration: 01/05/2016 – 03/04/2016.
• Late registration: 03/05/2016 – 03/20/2016
• Cost per test:
– \$96/test
• Can apply for a Cy-Hope scholarship (reduction of \$25/test, up to 3
tests).
• Pick up applications at Counselors’ Corner or at Mr. Hernandez’s
office or Ms. Lewis’ office.
– \$11/test (free/reduced lunch program)
• Not eligible for Cy-Hope scholarship.
Confidence
Intervals
0 - 100
• Name my age
– within 10 years.
– within 5 years.
– within 1 year.
• Shoot the ball at a large trash can and make
• Shoot the ball at a carnival game of chance
What happens to your
confidence as the interval
gets smaller?
As the interval gets smaller,
down.
the wider the interval.
Point Estimate
• Use a single statistic based on
sample data to estimate a
population parameter
• Simplest approach
• But not always very precise due to
variation in the sampling
distribution
Confidence intervals
• Are used to estimate the
unknown population mean
• Formula:
estimate + margin of error
Margin of error
• Shows how accurate we believe our estimate
is
• The smaller the margin of error, the more
precise our estimate of the true parameter
• Formula:
 critical
m  
 value
  standard deviation
  
  of the statistic



Confidence level
• Is the success rate of the method
used to construct the interval
• “Using this method, ____% of the
time the intervals constructed will
contain the true population
parameter.”
What does it mean to be 95%
confident?
• 95% chance that m is contained in
the confidence interval. ( ? ? ? )
• The probability that the interval
contains m is 95%. ( ? ? ? )
• The method used to construct the
interval will produce intervals that
contain m 95% of the time. ( ? ? ? )
• Which is correct?
AP Statistics
Monday, 01 February 2016
• OBJECTIVE TSW determine confidence intervals.
• ASSIGNMENT DUE DATES
– WS Confidence Intervals #1
– WS Confidence Intervals #2
 tomorrow, 02/02/16
 Wednesday, 02/03/16
• Sampling Distributions tests are not graded.
• AP EXAM REGISTRATION (through 03/04/16)
www.TotalRegistration.net/AP/443381
Critical value (z*)
• Found from the confidence level
• The upper z-score with probability p lying to
its right under the standard normal curve
Confidence level
90%
95%
99%
z*=1.645
tail area z*=1.96
z*=2.576z*
.05
.025
.005
1.645
.05
.025 1.96
.005
2.576
Confidence interval for a
population mean (Formula):
Standard
Critical
value
deviation of the
statistic
  
x  z *

 n
estimate
Margin of error
Steps for doing a confidence
interval:
1) State the assumptions –
•
•
SRS taken from population
Sampling distribution is normal (or approximately
normal)
•
•
•
•
Given (normal)
Large sample size (approximately normal)
Graph data (approximately normal)
 is known
2) Calculate the interval
3) Write a statement about the interval in the
context of the problem (complete sentence).
Statement: (memorize!!)
We are ________% confident
that the true mean of context
lies within the interval ______
and ______.
Example 1: A test for the level of potassium
in the blood is not perfectly precise.
Suppose that repeated measurements for
the same person on different days vary
normally with  = 0.2. A random sample of
three has a mean of 3.2. What is a 90%
confidence interval for the mean potassium
level?
Assumptions:
1) Have an SRS of blood measurements
2) Potassium level is normally distributed (given)
3)  known
 0.2 
3.2  1.645 
  3.0101, 3.3899 
 3
We are 90% confident that the true mean
potassium level is between 3.010 and 3.390.
95% confidence interval?
Assumptions:
1) Have an SRS of blood measurements
2) Potassium level is normally distributed
(given)
3)  known
 0.2 
3.2  1.96 
  2.9737, 3.4263
 3
We are 95% confident that the true mean
potassium level is between 2.974 and
3.426.
99% confidence interval?
Assumptions:
1) Have an SRS of blood measurements
2) Potassium level is normally distributed
(given)
3)  known
 0.2 
3.2  2.576 
  2.9026,3.4974 
 3
We are 99% confident that the true mean
potassium level is between 2.903 and 3.497.
What happens to the interval as the confidence
level increases?
the interval gets wider as the
confidence level increases
How can you make the margin of
error smaller?
• z* smaller
(lower confidence level)
•  smaller
(less variation in the population)
• n larger
Really cannot
(to cut the margin of error
in half, n must
change!
be 4 times as big)
Example 2: A random sample of 50
JVHS students was taken and their
mean SAT score was 1250. (Assume
 = 105) What is a 95% confidence
interval for the mean SAT scores of
JVHS students?
We are 95% confident that the true
mean SAT score for JVHS students
is between 1220.9 and 1279.1
How do you find a critical value (z*) for
a given confidence level?
• Use invNorm on the calculator.
– Example: For a 90% confidence level,
invNorm(0.95) = 1.644853626 . . .
–
For an 84% confidence level,
invNorm(0.92) = 1.405071561 . . .
Example 2.5:
Suppose that we have this random sample of
SAT scores for JVHS students:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for the true
mean SAT score? (Assume  = 105)
Assumptions:
•SRS (given)
•The distribution is approximately normal (“boxplot is symmetrical”
or “quantile plot is linear”).
•σ is given.
 105 
1192.857  1.96 

 7
We are 95% confident that the true mean SAT score for
JVHS students is between 1115.072 and 1270.642.
AP Statistics
Tuesday, 02 February 2016
• OBJECTIVE TSW (1) finish viewing the presentation
on confidence intervals, (2) turn in WS #1, and (3)
work on WS #2.
• ASSIGNMENTS DUE
– WS Confidence Intervals #1  wire basket
– WS Test Addendum: Continuous Distributions
tray
• ASSIGNMENT DUE TOMORROW
– WS Confidence Intervals #2
• QUIZ: Confidence Intervals is tomorrow.
 black
Finding a sample size:
• If a certain margin of error is wanted, then to
find the sample size necessary for that
margin of error use:
 
m  z *

 n
Always round up to the nearest person!
Example 3: The heights of JVHS
male students is normally distributed
with  = 2.5 inches. How large a
sample is necessary to be accurate
within + 0.75 inches with a 95%
confidence interval?
2.5 

Solve 0.75  1.96 
 for n.
 n
n = 43
Example 4: In a randomized comparative
experiment on the effects of calcium on blood
pressure, researchers divided 54 healthy, white
males at random into two groups, giving them
either calcium or a placebo. The paper reports
a mean seated systolic blood pressure of 114.9
with standard deviation of 9.3 for the placebo
group. Assume systolic blood pressure is
normally distributed.
Can you find a z-interval for this problem?
Why or why not?
No – the population standard deviation (σ) is not known.
Student’s t- distribution
• Developed by William Gosset
• Continuous distribution
• Unimodal, symmetrical, bell-shaped density
curve
• Above the horizontal axis
• Area under the curve equals 1
• Based on degrees of freedom
How does t compare to
normal?
• Shorter &amp; more spread out
• More area under the tails
• As n increases, t-distributions
become more like a standard
normal distribution
How to find t*
Can also use invT on the calculator!
• Use Table B for t distributions (green chart)
t* value level
with 5%
is above
• Need
Lookupper
up confidence
at bottom
&amp; df– on
the sides so 95% is below
• df = n – 1
invT(p,df)
Find these t*
90% confidence when n = 5 t* =2.132
95% confidence when n = 15 t* =2.145
Assumptions for t-inference
• Have an SRS from population
•  unknown
• Normal distribution
– Given
– Large sample size
– Check graph of data
Formula:
Standard
deviation of
Critical value
statistic
Confidence Interval:
 s 
x t * 

 n
estimate
Margin of error
For Ex. 4: Find a 95% confidence
interval for the true mean systolic
blood pressure of the placebo group.
Assumptions:
• Have an SRS of healthy, white males
• Systolic blood pressure is normally distributed
(given).
•  is unknown
t *  invT  0.975, 27  1  2.056
 9.3 
95% CI  114.9  2.056 
  (111.220, 118.580)
 27 
We are 95% confident that the true mean systolic
blood pressure is between 111.220 and 118.580.
Robust
• An inference procedure is ROBUST if the
confidence level or p-value doesn’t change
much if the assumptions are violated.
• t-procedures can be used with some
skewness, as long as there are no outliers.
• Larger n can have more skewness.
Example 5: A medical researcher measured
the pulse rate of a random sample of 20 adults
and found a mean pulse rate of 72.69 beats per
minute with a standard deviation of 3.86 beats
per minute. Assume pulse rate is normally
distributed. Compute a 95% confidence interval
for the true mean pulse rates of adults. (Just
find the interval.)
t *  invT  0.975, 20  1  2.0930
 3.86 
95% CI  72.69  2.0930 

 20 
(70.8834, 74.4965)
Another medical researcher claims that the
true mean pulse rate for adults is 72 beats per
minute. Does the evidence support or refute
this? Explain.
The 95% confidence interval contains
the claim of 72 beats per minute.
Therefore, there is no evidence to doubt
the claim.
Example 6: Consumer Reports tested 14
randomly selected brands of vanilla yogurt and
found the following numbers of calories per
serving:
160 200 220 230 120 180 140
130 170 190 80 120 100 170
Compute a 98% confidence interval for the
average calorie content per serving of vanilla
yogurt. (Just find the interval.)
t *  invT  0.99, 14  1  2.6503
x  157.8571
 44.7521
s  44.7521
98% CI  157.8571  2.6503 

14 

(126.1618, 189.5524)
A diet guide claims that you will get 120
calories from a serving of vanilla yogurt.
Note:
confidence
intervals
tell us
What
does
this evidence
indicate?
if something is NOT EQUAL –
never less or greater than!
Since 120 calories is not contained
within the 98% confidence interval, the
evidence suggests that the average
calories per serving does not equal 120
calories.
Some Cautions:
• The data MUST be a SRS from the
population
• The formula is not correct for more
complex sampling designs, i.e.,
stratified, etc.
• No way to correct for bias in data
Some Cautions (continued):
• Outliers can have a large effect on
confidence interval
• Must know  to do a z-interval –
which is unrealistic in practice
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