chapter 4 (part 1)

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Transcript chapter 4 (part 1)

ROHANA BINTI ABDUL HAMID
INSTITUTE FOR ENGINEERING MATHEMATICS (IMK)
UNIVERSITI MALAYSIA PERLIS
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CHAPTER 4
(PART 1)
STATISTICAL
INFERENCES
2
1. Introduction
2.Parameter
estimation
3.Hypothesis
testing
Confidence
interval for
population
mean
Hypothesis
testing for
population
mean
Confidence
interval for
population
proportion
Hypothesis
testing for
population
proportion
3
4.1 INTRODUCTION
The field of statistical inference consist of those
methods used to make decisions or to draw
conclusions about a population.
These methods utilize the information contained in a
sample from the population in drawing conclusions.
Statistical Inference may be divided into two major
areas:
a) parameter estimation and
b) hypothesis testing.
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4.2 PARAMETER ESTIMATION
Definition 4.1: An Interval Estimate
In interval estimation, an interval is
constructed around the point estimate and it
is stated that this interval is likely to contain
the corresponding population parameter.
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Definition 4.2: Confidence Level and
Confidence Interval
Each interval is constructed with regard to a given
confidence level and is called a confidence
interval.
The confidence level associated with a confidence
interval states how much confidence we have that
this interval contains the true population parameter.
The confidence level is denoted by
1   100%
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4.2.1 Confidence Interval Estimates for
Population Mean
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EXAMPLE 4.1
SOLUTION
Z
2
Z   Z 0.025  1.96
10
2
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EXAMPLE 4.2
A publishing company has just published a new
textbook. Before the company decides the price at
which to sell this textbook, it wants to know the
average price of all such textbooks in the market.
The research department at the company took a
sample of 36 comparable textbooks and collected
the information on their prices. This information
produced a mean price RM 70.50 for this sample. It
is known that the standard deviation of the prices of
all such textbooks is RM4.50.
Construct a 90% confidence interval for the mean
price of all such college textbooks.
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Problem
statement
• The company wants to know the average price of
all such textbooks in the market.
• a sample of 36 comparable textbooks
and collected the information on their
prices.
Gather
information
• This information produced a mean price
RM 70.50 for this sample.
• It is known that the standard deviation
of the prices of all such textbooks is
RM4.50
Construct
confidence
interval
• Construct a 90% confidence interval
for the mean price of all such college
textbooks.
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SOLUTION
  1  0.9  0.1
Z   Z 0.1  Z 0.05  1.65
2
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4.2.2 The (1   )100% Confidence Interval Estimates
on the Differences between Two Population Means, 1  2
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1  2  0
( there is no difference
between the two population
mean)
If the confidence interval includes 0 we can
say that there is no significant difference
between the means of the two populations,
at a given level of confidence.
EXAMPLE 4.3
The scientist wondered whether there was a difference in the
average daily intakes of dairy products between men and
women. He took a sample of n=50 adult women and recorded
their daily intakes of dairy products in grams per day.
He did the same for adult men. A summary of his sample results
is listed below.
Sample size
Sample mean
Sample standard
deviation
Men
Women
50
780 grams per day
35
50
762 grams per day
30
Construct a 95% confidence interval for the difference in the
average daily intakes of daily products for men and women.
Can you conclude that there is a difference in the average
daily intakes of daily products for men and women?
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SOLUTION
 s2 s 2 
x1  x2   z  1  2 
n1 n2
2

 352 30 2
 780  762  1.96

 50 50

 18 12.78




 [5.22,30.78]
Since the value of 0 is not included in the
interval, we can conclude that at 95% confidence
interval there is a difference in the average daily
intake of daily products for men and women.
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EXERCISE 1
A study was carried out to estimate the mean
height in centimeters of 15 years old students in
Tunku Abdul Rahman Secondary School.
Previous studies indicate that the variance of
the height of such students is 80cm. A random
sample of 100 students was selected. Suppose
that the mean sample was x  145cm.
Find a 95% confidence interval for the mean
height of all 15 years old students in the school.
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EXERCISE 2
The average zinc concentration recovered
from a sample of zinc measurements in 36
different locations is found to be 2.6 grams per
millimeter.
Find a 95% confidence interval for the mean
zinc concentration in the river. Assume that
the population standard deviation is 0.3
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EXERCISE 3
The contents of 7 similar containers of sulfuric
acid are 9.8, 10.2 10.4, 9.8,10.0, 10.2 and 9.6
liters.
Find a 99% confidence interval for the mean
of all such containers, assuming an
approximate normal distribution.
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EXERCISE 4
An experiment was conducted with two types of
engines, A and B . Gas mileages in miles per gallon
were measured. Fifty experiments were conducted
using engine Type A and 75 experiments were done
for Engine Type B. The gasoline used and other
conditions were held constant. The average gas
mileage for Engine A was 42 miles per gallon and that
for Engine B was 36 miles per gallon.
Find a 96% confidence interval for the difference
between the gas mileages for the two types of
engines. Assume that population standard deviations
of gas mileages are 6 and 8 for engines A and B
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