Transcript Confidence Intervals of the Mean

Confidence Intervals of
the Mean
2nd part of ‘estimate of the
mean’ presentation
Mu is between the two values calculated by:
X  Z * σ/ n
X  Z * σ/ n
Always draw the N.D.
The shaded area can help us to see what
we mean by X  Z * σ/ n
It is the border of the shaded area to the right of the mean .
We are saying that the mean lies between that border and the
corresponding left-side border
Write the equation


X  Z * σ/ n 
μ  X  Z * σ/
Mu lies between the two values
within the parentheses
n

Practical Statement of Result
• With 95% confidence, we can say that u
will be ≤ 1.96 *S.E. and
• ≥ -1.96 * S.E.
• We call 1.96 the reliability coefficient
The Math
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X = 22,
n = 10,
б2 = 45
Review what the symbols mean?
Find the quantity Z * б / Γn
б = Γ45 = 6.7
б / Γn = 6.7/ Γ10 = 6.7 / 3.16 = 2.12
1.96 * 2.12 = 4.16
Mu is between 22 – 4.16 and 22 + 4.16
• 17.84 ≤ µ ≤ 26.16 with 95% confidence
‘Significance Level’
• Related to confidence interval
• 95% confidence level can be expressed as
0.05 significance level
• Alpha is term for significance level
Another Example
• Page 230, #13
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Take out diskette from back of book
Insert into computer
Click on Install
Check ASCII, excel, SPSS
Install to hard drive
Excel
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Go to “exercise”
Find “lowbwt”
Save to hard drive to work on
File will probably be gone next time you
return. Save the data set we want onto
your own floppy
Do Problem 13, just the males
Separate male & female,
how?
Important Statements in the
Problem
1. Large sample
Applications
• If we know a population mean & st. dev.,
we can calculate the probability that any
sample will have a stated mean.
• A certain large human pop’n has a cranial
length that is approx’ly normally distributed
with mean 185.6 mm and б of 12.7 mm.
µ = 185.6 mm
б = 12.7 mm
• What is the probability that a random
sample of size 10 from this population will
have a mean greater than 190?
• We can calculate this probability but why
would we?
Usefulness??
• Let’s say that it is accepted knowledge
that the population has a certain mean.
• I am working with a group of people.
• I want to know if they fit into this
population with regard to the particular
parameter. If the probability of the mean
of the sample is very low, perhaps it is not
really from the same population
Education Example
• Third-graders in the U.S. have an average
reading score of 124.
• Third-graders in a particular school have a
mean reading score of 120. What’s the
probability that they are from the same
population?
Back to Cranial Length
• µ = 185.6 mm б = 12.7 mm
• random sample of size 10 from this
population will have a mean greater
than 190?
• Have to find how far 190 is from 185.6
in units of standard error of the mean
Z  (190 185.6)
12.7/ 10
Probability of Mean of 190
Did you draw a normal
dist???
Z = 4.4
/ 12.7 / 3.16
Z = 1.09
0.138
Area = 0.138
185.6 190
The probability is 13.8%
0
1.09
The mean & st. dev. of serum iron values are
120 & 15 micrograms per 100 ml. What is the
probability that a random sample of 50 normal
men will yield a mean between 115 & 125
µg/100ml?
µ = 120
б = 15
Z1 = (115-120) / 15 / sqrt of 50
Z2 = (125-120) / 15 / sqrt of 50
Z1 = (115-120) / 15 / sqrt of 50
Z2 = (125-120) / 15 / sqrt of 50
Draw the Normal Distribution