Transcript Statistical Reasoning

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
8.2 Estimating Population Means
CONFIDENCE INTERVAL→
Range of values that is likely to contain the
value of the population mean. Written as a
compound inequality. { low < μ < high }
MARGIN OF ERROR→
Range of values likely to contain the population
parameter.
???HOW CONFIDENT ARE YOU???
1.96??
8.2 Estimating Population Means
For the purpose of Intro to Probability and
Statistics we will use a 95% CONFIDENCE
INTERVAL TO ESTIMATE THE POPULATION
MEAN → Using the margin of error and the sample
mean. The process is as follows.
2s
Margin of error (for 95% confidence) = E 
n
Confidence Interval = {x  E    x  E}  {x  E}
RECALL: NOTATION AND SYMBOL REVIEW:
n…Sample size
μ…Population mean
sx…Sample standard deviation
…Sample mean
x
8.2 Estimating Population Means
EXAMPLE:
A study conducted by the Garbage Project at the University of Arizona
involved a sample of 62 households; the households ranged in size from
2 to 11 members. The mean for the sample was 27.4 pounds of garbage
per household per week and the standard deviation was 12.5 pounds.
What is the population and estimate the population mean. What
conclusions can we make?
2s 2(12.5)
m.o.e. (for 95% confidence) = E 

 3.2
n
62
Confidence Interval = {x  E    x  E}  {x  E}
Confidence Interval = {27.4  3.2}  {24.2    30.6}
Therefore, the population of Arizona which is an appropriate cluster
sample for the USA uses with 95% confidence between 24.2 and 30.6
pounds of garbage per household per week.
8.2 Estimating Population Means

How do we choose the correct sample size?
WE CHOOSE OUR SAMPLE SIZE DEPENDENT
ON MARGIN OF ERROR
•In other words, normally we know what our acceptable
margin of error should be, thus using a little bit of
algebra we will choose the correct sample size.
CHOOSING THE CORRECT SAMPLE SIZE:
F
I
G
HJ
K
2
2s
2s
2s
If E 
 E n  2s  n 
n
then if
E
E
n
we would like to draw a sample from the population we will
use the population standard deviation. Therefore, sample
F2 I
size should be n  G J. [Where  maybe estimated]
HE K
2
8.2 Estimating Population Means

How do we choose the correct sample size?
EXAMPLE
CHOOSING THE CORRECT SAMPLE SIZE:
1
Imagine our m.o.e. should be within 0.01, and the  = .
4
 2 
Therefore, our sample size should be n  
 .
 E 
2
2
 1 
2
2( ) 

2
 2 
 0.5 
4
Hence, n  



50
  





 E 
 0.01 
 0.01 


our sample should be greater than or equal to 2500 subjects.
2
8.2 Estimating Population Means


Class work:
pg 353 # 1-25 all
?
95
Confident