CABT SHS Statistics & Probability - Estimation of Parameters (intro)

CABT Statistics & Probability – Grade 11 Lecture Presentation

The session shall begin

Estimation of Parameters
A CABT Grade 11 Statistics
and Probability Lecture

Inferential Statistics
Inferential statistics is concerned with
drawing conclusions and/or making decisions
concerning a population based only on sample
data.
Main functions of inferential
statistics:1. estimate population parameters
2. test statistical hypotheses

http://www.gohomeworkhelp.com/admin/photos/what-is-inferential-statistics.jpg
Inferential Statistics

Parameter & Statistic
A parameter is a descriptive measure
that describes a population.
A statistic is a descriptive measure
that describes a sample.
Usually, parameters are denoted by
lower-case GREEK letters (e.g.  or ),
while statistics use lower-case ROMAN
letter (e.g. x and s).

Estimation of Population Parameters
An estimator of a population
parameter is a random variable that
depends on sample information whose
value provides an approximation to this
unknown parameter.
A specific value of that random
variable is called an estimate.

Properties of Good Estimators
1. UNBIASED. The expected value or the mean of
the estimates obtained from samples of a given size
is equal to the parameter being estimated.
2. CONSISTENT. As sample size increases, the
value of the estimator approaches the value of the
parameter being estimated.
3. RELATIVELY EFFICIENT. Of all the statistics that
can be used to estimate a parameter, the relatively
efficient estimator has the smallest variance.

There are two types of estimates:
1. Point estimate: It is a specific
numerical value used to approximate a
population parameter.
2. Interval estimate: It is a range of
values used to approximate a
population parameter. It’s also called
a confidence interval.

Point Estimation
Point estimation is
the process of finding a
point estimate from a
random sample of a
population to
approximate a
parameter value. The
statistic value that
approximates a
parameter value is CABT Statistics & Probability – Grade 11 Lecture Presentation

The point
estimate is the
BEST GUESS or
the BEST
ESTIMATE of an
unknown
(fixed or random)
population
parameter.
Point Estimation

MEASURE
Population Value
(PARAMETER)
Sample Statistic
(POINT ESTIMATE)
Mean 
Standard
deviation  s
Proportion p
x
ˆp
Point Estimation

Notes:
1. Don’t expect that the point estimate is
exactly equal to the population
parameter.
2. Any point estimate used should be as
close as possible to the true
parameter.
3. Sampling should be done at random,
using a sample size that is as large
Point Estimation

The following are some situations
that use point estimates:
Point Estimation
a. (estimating a mean) A sample of 50
households is used to determine the average
number of children in a household in a
barangay.
b. (estimating a proportion) A sample of 50
households is used to determine the
percentage of households in a barangay
watching a particular teleserye.

The SAMPLE MEAN is used to
estimate the population mean .
x
The following are the lengths of seedlings in a plant box.
We want to estimate the mean length of the seedlings.
Point Estimation
(Exercise 2 of the textbook)

Estimate the mean length using the following:
a) average of the row averages
b) average of the column averages
c) using the average of the first row
d) using the average of the last two columns
Point Estimation
(Exercise 2 of the textbook)

To determine the average monthly income of
factory workers of a CEPZ company, ten
workers were randomly sampled. Their monthly
incomes (in thousand pesos) are shown in the
table. Calculate the point estimate for the
average monthly income.
CABT Statistics & Probability – Grade 11 Lecture PresentationCABT Statistics & Probability – Grade 11 Lecture Presentation
Point Estimation
Worker Monthly Income
(thousand pesos)
Worker Monthly Income
(thousand pesos)
1 11.5 6 11.5
2 10 7 12
3 9.5 8 10.5
4 9 9 11.5
5 10 10 9

Find the point estimate of the proportion
of private school teachers who are LET
passers in a city given that 480 out of a
sample of 600 randomly selected
teachers passed the LET.
Point Estimation

Find the point estimate of the proportion
of the number of junior high school
students who owns at least one cell
phone given the following sample:
Point Estimation
Grade
Number of
students surveyed
Number of
students surveyed
with at least one
cell phone
7 10 9
8 15 11
9 25 16
10 20 14

An interval estimate is a range of values
used to approximate a population
parameter. This estimate may or may not
contain the actual value of the parameter
being estimated.
Interval Estimation
An interval estimate has two components:
1. a range or interval of values
2. an associated level of confidence

Interval Estimation
Why use an interval
estimate instead?
• Using a point estimate, while unbiased,
poses a degree of uncertainty. There
is no way of expressing the degree of
accuracy of a point estimate.
• An interval estimate provides more
information about a population
characteristic than does a point

confidence n. a feeling or belief that
you can do something well or succeed
at something
(http://www.merriam-webster.com/dictionary/confidence )
Confidence Levels and Intervals

The confidence
level c of an
interval estimate is
the probability that
the parameter is
contained in the
interval estimate.
Confidence Levels

The value of c is given by
Confidence Levels
 1
where  represents a level
of significance, which
indicates the long-run
percentage of confidence
intervals which would include
the parameter being
estimated.
The value of the level of
significance  is always

The significance of the level of
significance
Confidence Levels
The level of significance  represents a
probability of lack of confidence; that is, the
probability of NOT capturing the value of a
population parameter in the interval
estimate.
The confidence level c = 1   , meanwhile
represents the probability of confidence
that the population parameter lies within

The significance of the level of
significance
Confidence Levels
 1

z
probability that 
lies in the interval
estimate
probability that 
does NOT lie in
the interval
estimate

A confidence interval is a specific
interval estimate of a parameter
determined by using data obtained from a
sample and by using the specific
confidence level of the estimate.
Confidence Intervals
http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests:-confidence-intervals-and-confidence-levels

Notes:
1. For a parameter , if P(a <  < b) = 1  
, then the interval a <  < b is called a
100(1  )% confidence interval of
.
2. In repeated samples of the population,
the true value of the parameter  would
be contained in 100(1  )% of intervals
calculated this way.

 1

2

REGION OF CONFIDENCE
100(1 - )% of all intervals contain the value of 

2
ˆ
Distribution ofˆ 's

Illustration:
A 95% confidence interval of
a population mean  means
that 95% of the samples from
the same population will
produce the same confidence
intervals that contain the
value of .
http://www.statistica.com.au/confidence_interval.html
Also, this means that
1 0.95 
so is the level
of significance.
0.05 

Illustration:
a 95% confidence interval for the mean  in a normally-distributed population

Determine the confidence level for the
following levels of significance:
Level of
Significance
Confidence
Level
  0.10       1 1 0.10 0.90 90%c
  0.25       1 1 0.25 0.75 75%c
  0.36       1 1 0.36 0.64 64%c

Determine the levels of significance for
the following confidence levels:
Confidence
Level
Level of
Significance
96%      1 1 0.96 0.04c
87%      1 1 0.87 0.13c
90%      1 1 0.90 0.10c

Point
Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Margin of Error Margin of Error
Width of
confidence interval
Important parts of a confidence interval

General Formula for Confidence
Intervals
The general formula for all confidence
intervals is given by:
The value of the reliability factor depends
on the desired level of confidence.
Point Reliability Standard
Estimate Factor Error
    
    
    
Wow!

Intervals
https://onlinecourses.science.psu.edu/stat504/sites/onlinecourses.science.psu.edu.stat504/files/lesson01/simple_expres_CI.gif

Intervals
Usually, the general formula for a confidence
interval
is written as where is the estimate of
the
parameter  and E is the margin of error.
ˆ E  ˆ
In INEQUALITY FORM, the confidence interval
of a parameter  is given by
ˆ ˆE E    

Population
Mean

Unknown
Confidence
Intervals
Population
Proportion

Known
Wow!

for Known and Unknown Variances

Confidence Intervals for
the Population Mean
To construct an interval estimate for
the population mean, we use
1.a point estimate for the mean.
2.a margin of error.

the Population Mean
The confidence interval for the
population mean  is given by
    x E x E
where E is the margin of error
dependent on a given confidence
level.
Wow!

the Population Mean
In the confidence interval
    x E x E
x E = lower confidence limit
x E = upper confidence limit
2E = width of the confidence interval
E = margin of error

the Population Mean
Population
Mean
Lower Confidence
Limit
Upper Confidence
Limit
Margin of Error Margin of Error
Width of Confidence Interval
x E x E
E E

the Population Mean
 1 
2

REGION OF CONFIDENCE
100(1 - )% of all intervals contain the value of the population
mean 

2
X
  100 1 %
x E x E

the Population Mean
Suppose that a sample is taken from a
normally-distributed population. If the sample
mean is 10, the confidence interval for the
population mean  at a margin of error of 2 is
    10 2 10 2 or   8 12
From the confidence interval, we have:
Lower confidence limit: 8
Upper confidence limit: 12
Width of confidence interval: 2E = 4 or 12 – 8 =
4

the Population Mean
Find the margin of error and the
width of the following confidence
intervals:Confidence
Interval
Width of
Confidence Interval
Margin of
Error
 5 3 2  
2
1
2
E  3 5
  2.5 4.3
 36.92 35.08 1.84  
1.84
0.92
2
E  35.08 36.92
 
1.8
0.9
2
E 4.3 2.5 1.8

the Population Mean
In constructing an interval estimate
for the population mean, we consider
two cases:
CASE 1 – the standard deviation  of
the population is known
CASE 2 – the standard deviation  of
the population is not known

the Population Mean
The standard deviation  of the
population is known.
A confidence interval for a population
mean  with a known standard
deviation  is based on the fact that
the sample means follow an
approximately normal distribution.

For the Population Mean
The Central Limit Theorem – A Throwback:
X
   X
n

 
The mean and standard deviation of the distribution
are, respectively,
If random samples of size n are drawn from a
population with replacement, then as n becomes
larger, the sampling distribution of the mean
approaches the normal distribution,
regardless of the shape of the population
distribution.

Because of the Central Limit Theorem, we can think of
the confidence level c = 1 –  as the area under the
standard normal curve between two CRITICAL
VALUES and .

2
z

2
z
 1

2

2

2
z 
2
z0

To get a 100(1 – )% confidence interval for a given level
of significance , we must include the central (1 – )
of the probability of the normal distribution, leaving a
total area of  in both tails, or /2 in each tail, of the
normal distribution.
  100 1 %

 1

2

2
X
x E x E

μμx

Intervals
extend from 100(1)%
of intervals
constructed
contain μ;
100()%
do not.
Sampling Distribution of the Mean
x E
x E
to
X
1x
2x
3x
nx
1nx
 1

2

2
M

Sampling Error
The difference between the point estimate
and the actual parameter value is called
the SAMPLING ERROR.
For the sampling distribution of sample
means, the sampling error is equal to
 x

Margin of Error
The margin of error E is the maximum
error of estimate given by
Wow!
 
2
X
E z or 
 
  
 2
E z
n
where  is the level of significance,  is the
population standard deviation, and n is the
sample size.

Steps in Constructing a Confidence Interval for a
Population Mean if the Standard Deviation is Known
STEP 1 – Calculate the sample mean. This is
the point estimate for the population mean .
STEP 2 – Find the z-score (critical value) that
corresponds to the confidence level .
STEP 3 – Calculate the margin of error E.
STEP 4 – Construct the confidence interval for :
    x E x E
Interpret the result.

Common Confidence Levels and the Corresponding z
Values
Confidence
Level
Confidence
Coefficient
c = 1 – 
Level of
Significance

Value of z-Value
80% 0.80 0.20 0.10 1.28
90% 0.90 0.10 0.05 1.645
95% 0.95 0.05 0.025 1.96
98% 0.98 0.02 0.01 2.33
99% 0.99 0.01 0.005 2.575
99.8% 0.998 0.002 0.001 3.08
99.9% 0.999 0.001 0.0005 3.27
2
z
2


A normally distributed
population has standard
deviation 1.5. A sample of
size 36 is obtained from the
population with sample mean
4. Find the margin of error for
a 99% confidence interval for
the population mean.

Solution
Given:     4, 1.5, 36, 0.99x n c
Value of :      1 1 0.99 0.01c
Value of z:   
2
0.005 2.575z z
Value of E:
 
   
     
   2
1.5
2.575 0.64
36
E z
n

Check your
understandingCompute the margin of
error for the estimation
of the population mean
 for a 90% confidence
with a sample of size
400 and population
standard deviation of
Mean and Variance of Sampling Distributions of Sample MeansEstimation of Parameters

A normally distributed population
has standard deviation 2. A
sample of size 25 is obtained from
the population with sample mean
10. Construct a confidence interval
for the mean  of the population
using
a. 90% confidence
b. 95% confidence

a. Solution
Given:     10, 2, 25, 0.90x n c
Value of :      1 1 0.90 0.10c
2
0.05 1.645z z
Value of E:  
   
     
   2
2
1.645 0.66
25
E z
n
Confidence limits:
  9.34 10.66
   
   
10 0.66 9.34
10 0.66 10.66
x E
x E
Confidence interval:

What does our answer mean
We are 90% confident
that the true
population mean  lies
between 9.34 and
10.66.
  9.34 10.66
Wow!

b. Solution
Given:     10, 2, 25, 0.95x n c
Value of :    1 0.95 0.05
2
0.025 1.96z z
   
     
   2
2
1.96 0.78
25
E z
n
Confidence limits:
  9.22 10.78
   
   
10 0.78 9.22
10 0.78 10.78
x E
x E

What does our answer mean
We are 95% confident
that the true
population mean  lies
between 9.22 and
10.78.
  9.22 10.78
Wow!

To determine the average amount
of purchase of its customers, a
convenience store samples 150 of
its customers. The average
purchase of the group is P 125. If
the store knew that the standard
deviation of all purchases is P 50,
what is the 95% confidence
interval for the average purchase

Solution
Given:
    125, 50, 150, 0.95x n c
Value of :    1 0.95 0.05
2
0.025 1.96z z
   
     
   2
50
1.96 8.00
150
E z
n

Solution
Confidence limits:
  117 133
   125 8 117x E
   125 8 133x E
We are 95% confident that the actual
average purchase is between P 117 and
P 133.
Conclusion:

A study of 400 kindergarten pupils
showed that they spend on average
5,000 hours watching TV. The
standard deviation of the population is
900.
a. Find the 95% confidence level of the
mean TV time for all pupils.
b. If a parent claimed that his children
watched 4,000 hours of TV, would
the claim be valid? Why?

a. Solution
Given:
    5000, 900, 400, 0.95x n c
Value of :    1 0.95 0.05
2
0.025 1.96z z
   
     
   2
900
1.96 88.2
400
E z
n

a. Solution (continued)
Confidence limits:
  4,911.8 5,088.2
   5000 88.2 4,911.8x E
   5,000 88.2 5,088.2x E
We are 95% confident that the actual
average TV time is between 4,911.8 and
5,088.2 hours.
Conclusion:

b.
Question: Is the claim of
the parent valid?
Answer: NO, the claim of the parent
is NOT valid because the average
is NOT in the confidence interval.

In a nutshell:
Steps in Finding the Confidence Interval for 
Given:     _____, _____, _____, _____x n c
Value of :    1 _____c
Value of z:  
2
_____z
Value of E: 
 
  
 2
E z
n
Confidence limits:   _____x E
  _____x E
  _____ _____

In a nutshell:
Steps in Finding the Confidence Interval for 
Conclusion:
We are _____%
confident that the true
mean / average _____ is
between _____ and
_____.
Okay!

Check your
understanding
Solve
Exercise 7(a)
and 8(a) on
page 166 of
your

Sample Size Determination
The MINIMUM sample size n needed to estimate
the population mean  is
where  is the level of significance,  is the
population standard deviation and E is the margin
of error.
  
  
 
 
2
2
z
n
E
Okay!

Sample Size Determination
Since the confidence interval widens as the
confidence level increases, the precision of
the interval estimate decreases. One way
to increase the precision without changing
c is to increase the sample size. The larger
the sample size, the better.
Why compute the sample
size?

Determine the minimum
sample size needed to
estimate the population
mean  with 95% confidence
using a margin of error of 4.
It is known that the
population standard

Solution:
Value of :    1 0.95 0.05
2
0.025 1.96z z
Minimum
sample size:
  
  
 
 
2
2
z
n
E
   
  
 
2
1.96 8
4
 15.37 16
Note:
ROUND UP
your answer
Given:    0.95, 4, 8c E

If the variance of a national
accounting examination is
900, how large a sample is
needed to estimate the true
mean score within 5 points
with 99% confidence?

Solution:
Value of :    1 0.99 0.01
2
0.005 2.575z z
Minimum
sample size:
  
  
 
 
2
2
z
n
E
   
  
 
2
2.575 30
5
 240 exams
Given:     0.99, 5, 900 30c E

Check your
understandingEhljie wants to conduct a
study on the average number
of hours a Grade 11 student
spends in studying Statistics
and Probability in a school
week with 98% confidence
and a margin of error of 2
hours. What sample size
should Ehljie use for her study
if the population standard
Okay!
Gamitin mo
‘yung
formula na
ibinigay ni
Sir!

CABT SHS Statistics & Probability - Estimation of Parameters (intro)

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CABT SHS Statistics & Probability - Estimation of Parameters (intro)