Download Lesson 28 Nov PS

Prob and Stats, Nov 28
Intro to Confidence Intervals
Book Sections: 8.1
Essential Questions: What is a confidence interval and how do I
compute one for the mean of a population?
Standards: PS-SPMJ,1, .4
Unbiased Estimators
• Sample means, variances, and proportions tend to target
the corresponding population parameters.
• We call these 3 statistics unbiased estimators.
• In other words, their sampling distributions have a mean
that is equal to the mean of the corresponding population
parameters.
• If we want to use a sample statistic (such as a
sample proportion) to estimate a population
parameter (such as the population proportion), it
is important that the sample statistic used as the
estimator targets the population parameter.
• Using a biased estimator might underestimate or
overestimate the population parameter.
• A good sample should be less variable and
unbiased.
• Variability – measured by the standard
error of the mean,  x ,which requires a
larger sample. (Bigger sample size, smaller
variation)
_
UNBIASED ESTIMATORS
point estimates
• These target the population parameters.
Mean x
2
Variance s
proportion pˆ
BIASED ESTIMATORS:
• These do NOT target the population parameter.
Median
Range
S tan dard Deviation s
• NOTE: The bias is relatively small when using s,
therefore, it is sometimes used to estimate the standard
deviation of the population.
Important Points
• Sampling mean is unbiased since it equals
the population mean.
• Each individual sample may not equal the
population mean, but it should be close.
• When we say 95% of the data is within 2
standard deviations we mean 95% of data
values are   2 .
Speaking specifically about the
mean…..
• When speaking of samples we mean that
95% of the intervals captured will contain
the population mean.
Definitions
• Interval Estimate – An interval bounded by two
values & used to estimate the value of a
population parameter. Statisticians prefer to use
an interval estimate rather than a point
estimate.
• Level of Confidence –(1   ); the proportion of all
interval estimates that involve the parameter being
estimated.
• Confidence Interval – An interval estimate with a
specified level of confidence.
What Does the Answer Mean?
• If the answer of a confidence interval is
(918.23, 930.23), and the confidence level
was 95%, then it means that the population
mean of the samples was captured 95% of
the time.
Remember…..
• Sample measures are called statistics.
• Population measures are called parameters.
Using 6, 8, 12, 16, 18, 21, 22, 25,
28, 29…….
• Find a point estimate
for
a. The mean
• Answer:
a. 18.5
b. Variance
b. s sq. = 64.06
c. St Deviation
c. s = 8.00
Confidence Interval for the
Mean – Pop. Standard
Deviation KNOWN
Z-Interval
Z Confidence Interval……
• To use, these conditions (assumptions) must be
met: When:
a. Sampling Distribution is normal
OR
b. Population standard deviation is known OR
the sample size is greater than or equal to
30. ( n  30 )
Z
• Z is a z-score around which the confidence
interval is built.
• We compute Z based on the invNorm
function with  2 as its argument.
Confidence Interval for Estimation of the Mean
Pt . Estimate  Confidence Coefficien t (St. Error of the Mean)
x  z (
2

n
)
This produces the lower and upper confidence levels.
Definition……
• Confidence Level: the probability that the
interval estimate will contain the parameter.
• The most common levels of confidence
are: 90%, 95%, 99%
Alpha ( )
• Alpha (  ) = the total area in both tails.
• Alpha/2 (  2 ) = the total area in one tail.
• Example:
Confidence Level = 90%
= 1 - .90 = .10
 = .10
 = .10/2 = 0.05
2
Z Alpha ( )
• The Z 
2
= Invnorm(1-  2 )
Example
• The president of a university wants to
estimate the average age of students. It is
known that the standard deviation is 2
years. A sample of 50 is selected and the
mean age is found to be 23.2 years. Find
the 95% confidence interval.
What does this answer mean?......
• “We can say with 95% confidence that the
average age of students at the university is
between 22.65 years and 23.75 years.”
Example
• A certain medication increases the pulse
rate. The variance is 25 beats/minute. A
sample of 30 users has an average rate of
104 beats/minute. Find the 99% confidence
interval.
Example
• A sample of 50 days showed a store served
an average of 182 customers. The
standard deviation was 8. Find the 90%
confidence interval.
How Wide are Various
Confidence Intervals?
• The higher the level of confidence, the
wider the interval.
Z / 2 Values
% Confidence
90
91
92
93
94
95
96
97
98
99
Z / 2
1.64
1.70
1.75
1.81
1.88
1.96
2.05
2.17
2.33
2.58
Classwork: Handout CW 11/28/16, 1-4
Homework – Due 11/29/16, 1-2