Probability and Statistics – Mrs. Leahy
Unit 8: Estimation
Section 1: Estimating µ when σ is known
Confidence Intervals for Means – Part 1
Recall: Due to time, money, etc. we usually don’t have access to all measurements of an entire population so
instead we rely on a ______________________.
New Terminology:
A _______________________ of a population parameter is an estimate of the parameter using a single number.
We will use ______ to represent the point estimate for the population mean µ .
The sample point estimate ____ and the mean ____ are usually not exactly equal.
The _________________________________ is the absolute value of the difference between the sample
point estimate and the true population parameter value: |𝑥̅ − 𝜇|
The reliability of an estimate is called the
___________________________________ and we use the variable “c”
You can choose c to be any value between 0 and 1 
usually use 0.90, 0.95, or 0.99
The value zc is called the ______________ value and is the number such that the
area under the standard normal curve between  zc and zc equals c.
Confidence Level and Critical Value
Symbolically:
Example 1:
Example 2: How would you actually calculate example 1?
Find a critical value z such that 99% of the area under the normal curve lies between 𝒛−.𝟗𝟗 and 𝒛.𝟗𝟗 .
From Unit 7, Section 3:
Work backwards from the table
Look for 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦
=
Find z (to two decimal places)
1−𝐴
2
Example 3: Find the critical value 𝑧𝑐 for c = 0.90.
Some Levels of Confidence and Their
Corresponding Critical Values:
An estimate is not very valuable unless we
know how “good” it is.
We can use the margin of error to
determine something called a “Confidence
Interval” for µ
Confidence Interval for µ:
an interval such that c is
the PERCENTAGE of all intervals generated by the same
process that contain µ.
Example 4:
(Come back to this after example 6)
Interpretation: We can conclude that with ______ confidence that the interval from _______ to _______ will
contain the POPULATION MEAN µ of the population of domesticated geese.
Example 5: Suppose that the standard deviation of all high school seniors’ SAT scores in a certain year was σ=150.
A random sample of 100 scores yielded a sample mean 𝑥̅ = 1010 . Let µ be the mean of all SAT scores in that
year. Find a 0.99 confidence interval for µ. Round your answers to integers.
Formulas:
(Come back to this after example 6)
Interpretation: We can conclude that with ______ confidence that the interval from _______ to _______ will
contain the mean SAT Score.
Example 6: Repeat example 5, this time using a 0.95 confidence interval.
(Come back to this)
Interpretation: We can conclude that with ______ confidence that the interval from _______ to _______ will
contain the mean SAT Score.
Interpretation of Confidence Intervals:
Once we have a specific confidence interval for µ, such
as 3 < µ < 5 , all we can say is that:
We are c% confident that the mean of the population µ
will be in that interval.
Section 2: Estimating µ when σ is unknown
Confidence Intervals for Means – Part 2
Situation 1:
Situation 2:
Want to estimate µ for population
Know σ of population
Know 𝑥̅ (average of sample of size n)
Use:
to find
Want to estimate µ for population
Don’t know σ of population
Know 𝑥̅ (average of sample of size n) and s (standard deviation of sample of size n)
Use: A non-normal distribution called The Student’s t distribution
The Student’s t-distribution
History Lesson: William S. Gossett, Statistician for Guinness Brewing Company
Gossett and other employees “discouraged” publication of research.
Gossett believed the research was important and published anyway under the name:
“Student”
Now instead of “Gossett’s t-distribution”, statistical literature refers to it as “The
Student’s t-distribution”
Assume that x has a normal
distribution with mean µ.
For samples of size n with sample
mean 𝑥̅ and sample standard
deviation s, the t variable
The t-distribution:
is ___________________ about the mean
of 0
is ___________________ with thicker
tails than the standard normal
distribution
has a Student’s t distribution with
degrees of freedom d.f. = n – 1
is dependent upon the degrees of
freedom:
as d.f. ________________, the
t-distribution approaches the standard normal distribution
Example 1:
Using a t-distribution table to find critical vales
Example 2:
Example 3: Find the critical value tc for a 0.90 confidence level for a t-distribution with sample size n=8.
Example 4: Find the values of t0.95 and t0.98 for a sample of size 5.
Confidence Interval for µ when σ is unknown
Example 5: A company has a new process for manufacturing large artificial sapphires. In a trial run, 37 sapphires
are produced. The mean weight for these 37 gems is 𝑥̅ =6.75 carats, and the sample standard deviation is s=0.33
carat. Let µ be the mean weight for the distribution of all sapphires produced by the new process.
Find a 95% confidence interval for µ.
Steps:
1. Find d.f.
2. Use t-table to find tc
3. Use formula to find E.
4. Use E to find interval.
5. Interpret interval.
Example 6: An archeologist discovers only seven fossil skeletons from a previously unknown species of miniature
horse. For this sample, the mean is 𝑥̅ = 46.14 and the sample standard deviation is 1.19. Find a 99% confidence
interval for the entire population of such horses.
Section 3: Estimating p in the Binomial Distribution
Confidence Intervals for the probability of success in a binomial distribution
Situation 3:
You have a situation with a binomial probability
distribution
You can assume that for a large enough number
of trials, the binomial distribution is close enough
to a normal distribution and can use this
information to estimate the value of p
Recall:
Binomial Probability Distribution
1. Fixed number of trials, n
2. Two possible outcomes, success & failure
3. Probability of success = p
4. Probability of failure = 1 – p
5. Looking for the Probability of r successes out of n trials
P(n,r)
Studies have shown that we get quite good answers as long as:
POINT ESTIMATES in the Binomial Case:
***POWERPOINT****
How to find a confidence interval for p
In a binomial experiment
Step 1: Find the point estimates 𝑝̂ , 𝑞̂
n trials
p=probability of success
Step 2: Verify that 𝑛𝑝̂ > 5 and 𝑛𝑞̂ > 5
q = 1 – p = probability of failure
r = number of successes out of n trials Step 3: Find the correct critical z value for your confidence level, c
Step 4: Calculate the Error
Step 5:
𝐸 ≈ 𝑧𝑐 √
CONFIDENCE INTERVAL:
̂−𝑬<𝒑<𝒑
̂+𝑬
𝒑
𝑝̂𝑞̂
𝑛
Example 1:
a) What is the number of trials n? What is the value of r?
b) What are the point estimates for p and q?
c) Is the number of trials large enough to justify a normal approximation to the binomial?
d) Find a 99% Confidence interval for p.
Example 2:
a)
b)
c)
d)
What is the point estimate for P?
Find a 90% confidence interval for p
Interpret the confidence interval.
Are we justified in using a normal approximation?
Sample Size for Estimating
Example 3:
Example 4: