Download Confidence Intervals

Warm Up 8/26/14
•
A study of college freshmen’s study habits found that the time (in hours) that
college freshmen use to study each week follows a distribution with a mean
of 7.2 hours and a standard deviation of 5.3 hours.
•
a. Calculate the probability that a randomly chosen freshman studies
more than 9 hours.
b. Find the probability that the average number of hours spent studying by
an SRS of 55 students is greater than 9 hours. Show your work.
c. What are the mean and standard deviation for the average number of
hours spent studying by an SRS of 55 freshmen?
How do I construct and
interpret Confidence
Intervals using the margin
of error?
Rate your confidence
0 - 100
• Shooting a basketball at a wading pool, will
make basket?
• Shooting the ball at a large trash can, will
make basket?
• Shooting the ball at a carnival, will make
basket?
What happens to your
confidence as the interval
gets smaller?
The larger your confidence,
the wider the interval.
Confidence Interval
Definition of a Confidence Interval
• An interval that is computed from sample data and
provides a range of plausible values for a population
parameter
• Is the success rate of the method used to construct
the interval
• Formula:
Mean of the sample + margin of error
Core Lesson
A sample of 150 teenagers finds the average amount of money
spent on music per month is $65.25. With 95% confidence, the
margin of error is calculated to be $4.50. Construct and
interpret the confidence interval.
Construct: 65.25 ± 4.50 produces an interval from 60.75 to 69.75
Interpret: We are 95% confident the interval from $60.75 to $69.75 contains the actual
mean amount of money teenagers spend on music per month.
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



Critical Values
• There are 4 typical levels of confidence:
99%, 98%, 95% and 90%.
Level of confidence
Z – Critical
Values
90%
1.645
95%
1.96
98%
2.33
99%
2.576
Confidence interval for a
population mean:
Standard
Critical
value
deviation of the
statistic
  
x  z *

 n
Estimate/
mean of
sample
Margin of error
Steps for doing a confidence
interval:
1) Assumptions –
•
•
SRS 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.
Statement (interpret):
(memorize!!)
We are ________% confident
that the true mean context lies
within the interval ______ and
______.
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:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
 known
 .2 
  3.0101, 3.3899 
3.2  1.645
 3
We are 90% confident that the true mean
potassium level is between 3.01 and 3.39.
95% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed
(given)
 known
 .2 
  2.9737, 3.4263
3.2  1.96
 3
We are 95% confident that the true mean
potassium level is between 2.97 and
3.43.
99% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed
(given)
 known
 .2 
3.2  2.576
  2.9026,3.4974
 3
We are 99% confident that the true mean
potassium level is between 2.90 and 3.50.
What happens to the interval as the
confidence level increases?
the interval gets wider as the
confidence level increases
A random sample of 50 CHHS
students was taken and their mean
SAT score was 1250. (Assume  =
105) What is a 95% confidence
interval for the mean SAT scores of
CHHS students?
We are 95% confident that the true
mean SAT score for CHS students is
between 1220.9 and 1279.1
Suppose that we have this random sample
of SAT scores:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for the
true mean SAT score? (Assume  = 105)
We are 95% confident that the true
mean SAT score for CHHS students is
between 1115.1 and 1270.6.
In a randomized comparative experiment
on the effects of calcium on blood
pressure, researchers divided 54 healthy
males at random into two groups, takes
calcium or 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.
For the Ex. 4: Find a 95% confidence
interval for the true mean systolic
blood pressure of the placebo group.
Assumptions:
• Have an SRS of healthy males
• Systolic blood pressure is normally distributed
(given).
•  is unknown
 9.3 
114.9  2.056
  (111.22, 118.58)
 27 
We are 95% confident that the true mean systolic
blood pressure is between 111.22 and 118.58.
Warm Up 8/27/14
Research has shown that replacement times for TV
sets have a mean of 8.2 years and a standard deviation
of 1.1 years. He randomly selects a sample of 50 TV
sets sold in the past and finds that the mean
replacement time is 7.8 years.
• (a) Find the probability that 40 randomly selected TV
sets will have mean replacement time of 7.8 years or
less.
Why is the Central Limit Theorem important
in statistics…
A sample of 300 high school students was
asked how many text messages they sent on a
given day. The mean was 38 with a standard
deviation of 17.5.
(a) If we had 49 such samples, what would we
expect the mean and standard deviation of the
sampling distribution of means to be?
Ex. 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.
(70.883, 74.497)
Ex. – 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.
(126.16, 189.56)
Find a sample size:
2
The heights of CHHS male students
is normally distributed with  = 2.5
inches. How large a sample is
necessary to be accurate within + .75
inches with a 95% confidence
interval?
n = 43