Steps to Construct ANY Confidence Interval:

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Steps to Construct ANY Confidence Interval
PANIC
P
Parameter of Interest (what are you looking for?)
A
Assumptions (what are the conditions?)
N
Name the type of interval (what type of data do
we have?)
I
Interval (Finally! You can calculate!)
C
Conclusion in context (I am ___ confident the
true parameter lies between ________ and
_________)
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Example 1 A news release by the IRS reported 90
of all Americans fill out their tax forms
correctly. A random sample of 1500 returns
revealed that 1200 of them were correctly filled
out. Calculate a 92 confidence interval for the
proportion of Americans who correctly fill out
their tax forms. Is the IRS correct in their
report?
P
The true percent of Americans who fill out their
tax forms correctly
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A
Says randomly selected
SRS
Normality
0.80
Yes, safe to assume an approximately normally
distribution
It is safe to assume that there are more than
15,000 people who file their taxes
Independence
4
N
One Sample Proportion Interval
I
Z ?
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Confidence Level (C) Upper tail prob. Z Value
92
0.04
0.04
0.04
0.92
Z?
Z?
6
Confidence Level (C) Upper tail prob. Z Value
92
? 1.75
0.04
0.04
0.04
0.92
Z?
Z?
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N
One Sample Proportion Interval
I
8
C
I am 92 confident the true percent of Americans
who fill out their tax forms correctly is between
78.19 and 81.8
Is the IRS correct in their report?
No,
90 is not in the interval!
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Sample size for a Desired Margin of Error If we
want the margin of error in a level C confidence
interval for p to be m, then we need n subjects
in the sample, where
p
An estimate for
Note If p is unknown use the most conservative
value of p 0.5. Since n is the sample size, it
must be a whole number!!! Round up!
n ?
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Example 2 You wish to estimate with 95
confidence the proportion of computers that need
repairs or have problems by the time the product
is three years old. Your estimate must be
accurate within 3.5 of the true proportion.
a. Find the sample size needed if a prior study
found that 19 of computers needed repairs or had
problems by the time the product as three years
old.
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Example 2 You wish to estimate with 95
confidence the proportion of computers that need
repairs or have problems by the time the product
is three years old. Your estimate must be
accurate within 3.5 of the true proportion.
b. If no preliminary estimate is available, find
the most conservative sample size required.
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Example 2 You wish to estimate with 95
confidence the proportion of computers that need
repairs or have problems by the time the product
is three years old. Your estimate must be
accurate within 3.5 of the true proportion.
c. Compare the results from a and b.
Using 0.5 makes the sample size very large,
ensuring that enough people will be surveyed.
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Confidence Interval for a Population Mean (?
known)
(Z-Interval)
estimate ? margin of error
estimate ? critical value ? standard error
?
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Properties of Confidence Intervals for Population
Mean

• The interval is always centered around the
  statistic

• The higher the confidence level, the wider the
  interval becomes

• If you increase n, then the margin of error
  decreases

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Calculator Tip
Z-Interval
Stat Tests ZInterval
Data If given actual values
Stats If given summary of values
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Interpreting a Confidence Interval
What you will say
I am C confident that the true parameter is
captured in the interval _____ to ______
What it means
If we took many, many, SRS from a population and
calculated a confidence interval for each sample,
C of the confidence intervals will contain the
true mean
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CAUTION!
Never Say
The interval will capture the true mean C of the
time.
It either does or does not!
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Conditions for a Z-Interval
(problem should say)

1. SRS

(CLT or population approx normal)
2. Normality
(Population 10x sample size)
3. Independence
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Steps to Construct ANY Confidence Interval
PANIC
P
Parameter of Interest (what are you looking for?)
A
Assumptions (what are the conditions?)
N
Name the type of interval (what type of data do
we have?)
I
Interval (Finally! You can calculate!)
C
Conclusion in context (I am ___ confident the
true parameter lies between ________ and
_________)
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Example 1 Serum Cholesterol-Dr. Paul Oswick
wants to estimate the true mean serum HDL
cholesterol for all of his 20-29 year old female
patients. He randomly selects 30 patients and
computes the sample mean to be 50.67. Assume
from past records, the population standard
deviation for the serum HDL cholesterol for 20-29
year old female patients is ?13.4.

1. Construct a 95 confidence interval for the mean
   serum HDL cholesterol for all of Dr. Oswicks
   20-29 year old female patients.

P
The true mean serum HDL cholesterol for all of
Dr. Oswicks 20-29 year old female patients.
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A
SRS
Says randomly selected
Normality
Approximately normal by the CLT (n ? 30)
I am assuming that Dr. Oswick has 300 patients or
more.
Independence
N
One sample Z-Interval
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I
23
C
I am 95 confident the true mean serum HDL
cholesterol for all of Dr. Oswicks 20-29 year
old female patients is between 45.875 and 55.465
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Example 1 Serum Cholesterol-Dr. Paul Oswick
wants to estimate the true mean serum HDL
cholesterol for all of his 20-29 year old female
patients. He randomly selects 30 patients and
computes the sample mean to be 50.67. Assume
from past records, the population standard
deviation for the serum HDL cholesterol for 20-29
year old female patients is ?13.4.
b. If the US National Center for Health
Statistics reports the mean serum HDL cholesterol
for females between 20-29 years old to be ? 53,
do Dr. Oswicks patients appear to have a
different serum level compared to the general
population? Explain.
No,
53 is contained in the interval.
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Example 1 Serum Cholesterol-Dr. Paul Oswick
wants to estimate the true mean serum HDL
cholesterol for all of his 20-29 year old female
patients. He randomly selects 30 patients and
computes the sample mean to be 50.67. Assume
from past records, the population standard
deviation for the serum HDL cholesterol for 20-29
year old female patients is ?13.4.
c. What two things could you do to decrease your
margin of error?
Increase n
Lower confidence level
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Example 2 Suppose your class is investigating
the weights of Snickers 1-ounce Fun-Size candy
bars to see if customers are getting full value
for their money. Assume that the weights are
Normally distributed with standard deviation
0.005 ounces. Several candy bars are randomly
selected and weighed with sensitive balances
borrowed from the physics lab. The weights are
0.95 1.02 0.98 0.97 1.05 1.01 0.98 1.00
ounces. Determine a 90 confidence interval
for the true mean, µ. Can you say that the bars
weigh 1oz on average?
P
The true mean weight of Snickers 1-oz Fun-size
candy bars
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A
Says randomly selected
SRS
Normality
Approximately normal because the population is
approximately normal
I am assuming that Snickers has 80 bars or more
in the 1-oz size
Independence
N
One sample Z-Interval
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I
29
C
I am 90 confident the true mean weight of
Snickers 1-oz Fun-size candy bars is between
.9921 and .9979 ounces. I am not confident that
the candy bars weigh as advertised at the 90
level.
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Choosing a Sample Size for a specific margin of
error
Note Always round up! You cant have part of a
person! Ex 163.2 rounds up to 164.
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Example 3 A statistician calculates a 95
confidence interval for the mean income of the
depositors at Bank of America, located in a
poverty stricken area. The confidence interval
is 18,201 to 21,799.

1. What is the sample mean income?

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Example 3 A statistician calculates a 95
confidence interval for the mean income of the
depositors at Bank of America, located in a
poverty stricken area. The confidence interval
is 18,201 to 21,799.
b. What is the margin of error?
m
m 21,799 20,000
m 1,799
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Example 4 A researcher wishes to estimate the
mean number of miles on four-year-old Saturn
SCIs. How many cars should be in a sample in
order to estimate the mean number of miles within
a margin of error of ? 1000 miles with 99
confidence assuming ?19,700.
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8.3 Estimating a Population Mean
In the previous examples, we made an unrealistic
assumption that the population standard deviation
was known and could be used to calculate
confidence intervals.
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Standard Error
When the standard deviation of a statistic is
estimated from the data
When we know ? we can use the Z-table to make a
confidence interval. But, when we dont know it,
then we have to use something else! (Calculator
Bingo activity p. 502)
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Properties of the t-distribution

• s is unknown
• Degrees of Freedom n 1
• More variable than the normal distribution (it
  has fatter tails than the normal curve)
• Approaches the normal distribution when the
  degrees of freedom are large (sample size is
  large).
• Area is found to the right of the t-value

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Properties of the t-distribution

• If n lt 15, if population is approx normal, then
  so is the sample distribution. If the data are
  clearly non-Normal or if outliers are present,
  dont use!

• If n gt 15, sample distribution is normal, except
  if population has outliers or strong skewness

• If n ? 30, sample distribution is normal, even
  if population has outliers or strong skewness

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Use invT on calculator Go to 2nd VARS - 4
invT Type in invT((1C)/2, n-1) Example 1
Suppose you want to construct a 90 confidence
interval for the mean of a Normal population
based on SRS of size 10. What critical value t
should you use? Degrees of freedom n 1
10-1 9 Calculate invT((1.90)/2, 9)
1.833 t 1.833
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Example 2 Practice finding t
n Degrees of Freedom (n-1) Confidence Interval t
n 10 99 CI
n 20 90 CI
n 40 95 CI
n 30 99 CI
9
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Example 2 Practice finding t
n Degrees of Freedom Confidence Interval t
n 10 99 CI
n 20 90 CI
n 40 95 CI
n 30 99 CI
9
3.250
19
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Example 2 Practice finding t
n Degrees of Freedom Confidence Interval t
n 10 99 CI
n 20 90 CI
n 40 95 CI
n 30 99 CI
9
3.250
19
1.729
39
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Example 2 Practice finding t
n Degrees of Freedom Confidence Interval t
n 10 99 CI
n 20 90 CI
n 40 95 CI
n 30 99 CI
9
3.250
19
1.729
39
2.042
29
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Example 2 Practice finding t
n Degrees of Freedom Confidence Interval t
n 10 99 CI
n 20 90 CI
n 40 95 CI
n 30 99 CI
9
3.250
19
1.729
39
2.042
29
2.756
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Calculator Tip
Finding P(t)
2nd Dist tcdf( lower bound, upper bound,
degrees of freedom)
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One-Sample t-interval
Calculator Tip
One sample t-Interval
Go to Stat Tests TInterval
Data If given actual values
Stats If given summary of values
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Conditions for a t-interval

1. SRS

(problem should say)
(population approx normal and nlt15, or moderate
size (15 n lt 30) with moderate skewness or
outliers, or large sample size n 30)
2. Normality
3. Independence
(Population 10x sample size)
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Robustness
The probability calculations remain fairly
accurate when a condition for use of the
procedure is violated
The t-distribution is robust for large n values,
mostly because as n increases, the t-distribution
approaches the Z-distribution. And by the CLT,
it is approx normal.
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Example 3 As part of your work in an
environmental awareness group, you want to
estimate the mean waste generated by American
adults. In a random sample of 20 American
adults, you find that the mean waste generated
per person per day is 4.3 pounds with a standard
deviation of 1.2 pounds. Calculate a 99
confidence interval for ? and explain its
meaning to someone who doesnt know statistics.
P
The true mean waste generated per person per day.
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A
Says randomly selected
SRS
Normality
15ltnlt30. We must assume the population doesnt
have strong skewness. Proceeding with caution!
It is safe to assume that there are more than 200
Americans that create waste.
Independence
N
One Sample t-interval
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I
df
20 1
19
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I
df 20 1 19
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C
I am 99 confident the true mean waste generated
per person per day is between 3.5323 and 5.0677
pounds.