This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
Detailed explanation about line graphs and frequency polygons with examples..
Frequency polygons in discrete series and in continuous series is also explained..
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Level : 11
Bibiliography: Statistics for economics class 11 textbook by Sandeep Garg
A presentation by Amrutha Varshini
Detailed explanation about line graphs and frequency polygons with examples..
Frequency polygons in discrete series and in continuous series is also explained..
This ppt explains about " How to draw Frequency polygons with the help of histograms as well as " How to draw Frequency polygons without the help of histograms step by step
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Bibiliography: Statistics for economics class 11 textbook by Sandeep Garg
A presentation by Amrutha Varshini
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
HW1_STAT206.pdfStatistical Inference II J. Lee Assignment.docxwilcockiris
HW1_STAT206.pdf
Statistical Inference II: J. Lee Assignment 1
Problem 1. Suppose the day after the Drexel-Northeastern basketball game, a poll of 1000 Drexel students
was conducted and it was determined that 850 out of the 1000 watched the game (live or on television).
Assume that this was a simple random sample and that the Drexel undergraduate population is 20000.
(a) Generate an unbiased estimate of the true proportion of Drexel undergraduate students who watched
the game.
(b) What is your estimated standard error for the proportion estimate in (a)?
(c) Give a 95% confidence interval for the true proportion of Drexel undergraduate students who watched
the game.
Problem 2. (Exercise 18 in Chapter 7 of Rice) From independent surveys of two populations, 90% con-
fidence intervals for the population means are conducted. What is the probability that neither interval
contains the respective population mean? That both do?
Problem 3. (Exercise 23 in Chapter 7 of Rice)
(a) Show that the standard error of an estimated proportion is largest when p = 1/2.
(b) Use this result and Corollary B of Section 7.3.2 (also, on Page 17 of the lecture notes) to conclude that
the quantity
1
2
√
N − n
N(n − 1)
is a conservative estimate of the standard error of p̂ no matter what the value of p may be.
(c) Use the central limit theorem to conclude that the interval
p̂ ±
√
N − n
N(n − 1)
contains p with probability at least .95.
HW2_STAT206.pdf
Statistical Inference II: J. Lee Assignment 2
Problem 1. The following data set represents the number of NBA games in January 2016, watched by 10
randomly selected student in STAT 206.
7, 0, 4, 2, 2, 1, 0, 1, 2, 3
(a) What is the sample mean?
(b) Calculate sample variance.
(c) Estimate the mean number of NBA games watched by a student in January 2016.
(d) Estimate the standard error of the estimated mean.
Problem 2. True or false? Tell me why for the false statements.
(a) The center of a 95% confidence interval for the population mean is a random variable.
(b) A 95% confidence interval for µ contains the sample mean with probability .95.
(c) A 95% confidence interval contains 95% of the population.
(d) Out of one hundred 95% confidence intervals for µ, 95 will contain µ.
Problem 3. An investigator quantifies her uncertainty about the estimate of a population mean by reporting
X ± sX . What size confidence interval is?
Problem 4. For a random sample of size n from a population of size N, consider the following as an
estimate of µ:
Xc =
n∑
i=1
ciXi,
where the ci are fixed numbers and X1, . . . ,Xn are the sample. Find a condition on the ci such that the
estimate is unbiased.
Problem 5. A sample of size 100 has the sample mean X = 10. Suppose the we know that the population
standard deviation σ = 5. Find a 95% confidence interval for the population mean µ.
Problem 6. Suppose the we know that the population standard deviation σ = 5. Then how large should a
sample be to estimate the popula.
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The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
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The Art Pastor's Guide to Sabbath | Steve Thomason
Statistik 1 7 estimasi & ci
1. l
Chapter Nine
Estimation and Confidence IInntteerrvvaallss
GOALS
1. Define a what is meant by a point estimate.
2. Define the term level of confidence.
3. Construct a confidence interval for the population
mean when the population standard deviation is
known.
4. Construct a confidence interval for the population
mean when the population standard deviation is
unknown.
5. Construct a confidence interval for the population
proportion.
6. Determine the sample size for attribute and variable
sampling.
2.
3. Point and Interval Estimates
A point estimate is a single value (statistic)
used to estimate a population value
(parameter).
A interval estimate is a range of values within
which the population parameter is expected to
occur.
The interval within which a population
parameter is expected to occur is called a
confidence interval.
The specified probability is called the level of
confidence.
The two confidence intervals that are used
extensively are the 95% and the 99%.
4. Interval Estimation for
The Population Mean
Is the Population Normal
Is n 30 or more ? Is the population standard
deviation known ?
Use a Non
Parametric Test
Use the Z
Distribution
Use the t
Distribution
Use the Z
Distribution
No
No No
Yes
Yes Yes
5. Confidence Intervals
The degree to which we can rely on the statistic is as
important as the initial calculation. Remember, most
of the time we are working from samples. And
samples are really estimates. Ultimately, we are
concerned with the accuracy of the estimate.
1. Confidence interval provides Range of Values
Based on Observations from 1 Sample
1. Confidence interval gives Information about
Closeness to Unknown Population Parameter
Stated in terms of Probability
Knowing Exact Closeness Requires Knowing
Unknown Population Parameter
6. Areas Under the Normal Curve
Between:
± 1 s - 68.26%
± 2 s - 95.44%
± 3 s - 99.74%
μ
If we draw an observation
from the normal distributed
population, the drawn value is
likely (a chance of 68.26%) to
lie inside the interval of
(μ-1σ, μ+1σ).
P((μ-1σ <x<μ+1σ) =0.6826.
μ-2σ μ+2σ
μ-3σ μ-1σ μ+1σ
μ+3σ
7. Elements of Confidence Interval
Estimation
A probability that the population parameter
falls somewhere within the interval.
Confidence Interval
Sample Statistic
Confidence Limit
(Lower)
Confidence Limit
(Upper)
8. Confidence Intervals
X ±Z×s = ± ×s
X Z n _
m -2.58×s m -1.645×s m m +1.645×s m +2.58×s
m -1.96×s m +1.96×s
90% Samples
95% Samples
99% Samples
sx
X
X
X X X X
X X
9. Level of Confidence
1. Probability that the unknown population
parameter falls within the interval
2. Denoted (1 - a) % = level of confidence
a Is the Probability That the Parameter Is
Not Within the Interval
1. Typical Values Are 99%, 95%, 90%
10. Interpreting Confidence Intervals
Once a confidence interval has been
constructed, it will either contain the
population mean or it will not.
For a 95% confidence interval, if you were to
produce all the possible confidence intervals
using each possible sample mean from the
population, 95% of these intervals would
contain the population mean.
11. Intervals & Level of Confidence
Sampling
Distribution
of Mean
_
a/2 1 - a a/2
Large Number of Intervals
Intervals
Extend from
(1 - a) % of
Intervals
Contain m .
a % Do Not.
m`x = m
X _
sx
- × s
to
X
X Z
X
X Z
+ ×
s
12. Point Estimates and Interval Estimates
X Z s
n
±( a /2)×s = X ±( Z
a /2)×
X
The factors that determine the width of a
confidence interval are:
1. The size of the sample (n) from which the
statistic is calculated.
2. The variability in the population, usually
estimated by s.
3. The desired level of confidence.
13. Point and Interval Estimates
If the population standard deviation is known
or the sample is greater than 30 we use the z
distribution.
X ± z s
n
14. CONTOH
Penelitian dilakukan untuk mengetahui
pendapatan bersih PKL di Surabaya. Dari 100
orang sampel random diketahui rata-rata
pendapatan bersih per hari PKL Rp 50.000
dengan simpangan baku RP 15.000.
Berdasarkan data tersebut lakukan estimasi
pendapatan bersih PKL di Surabaya dengan
tingkat keyakinan 95%.
15. Point and Interval Estimates
If the population standard deviation is unknown
and the sample is less than 30 we use the t
distribution.
X ±t s
n
16. Student’s t-Distribution
The t-distribution is a family of distributions
that is bell-shaped and symmetric like the
standard normal distribution but with greater
area in the tails. Each distribution in the t-family
is defined by its degrees of freedom.
As the degrees of freedom increase, the t-distribution
approaches the normal
distribution.
17. About Student
Student is a pen name for a statistician
named William S. Gosset who was not
allowed to publish under his real name.
Gosset assumed the pseudonym Student for
this purpose. Student’s t distribution is not
meant to reference anything regarding
college students.
19. Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
0 t
Student’s t Table
Assume:
n = 3
df = n - 1 = 2
a = .10
a/2 =.05
a / 2
t Values 2.920
.05
20. Degrees of freedom
Degrees of freedom refers to the number of
independent data values available to estimate
the population’s standard deviation. If k
parameters must be estimated before the
population’s standard deviation can be
calculated from a sample of size n, the
degrees of freedom are equal to n - k.
21. Degrees of Freedom (df )
1. Number of Observations that Are Free to Vary After
Sample Statistic Has Been Calculated
2. Example
Sum of 3 Numbers Is 6
X1
= 1 (or Any Number)
X2
= 2 (or Any Number)
X3
= 3 (Cannot Vary)
Sum = 6
degrees of freedom
= n -1
= 3 -1
= 2
22. t-Values
t = x -m
where:
= Sample mean
= Population mean
s = Sample standard deviation
n = Sample size
s
n
xm
23. Estimation Example
Mean (s Unknown)
X
A random sample of n = 25 has = 50 and S = 8.
Set up a 95% confidence interval estimate for m.
X t
S
n
X t
S
- a n - × £ m £ + a n -
×
n - × £ £ + ×
/ , / ,
. .
2 1 2 1
m
m
. £ £
.
50 2 0639
8
25
50 2 0639
8
25
46 69 53 30
24. Central Limit Theorem
For a population with a mean m and a variance s2
the sampling distribution of the means of all possible
samples of size n generated from the population will
be approximately normally distributed.
The mean of the sampling distribution equal to m and
the variance equal to s2/n.
The population
X ~ ?(m,s 2 )
distribution
The sample mean of n X ~ N( m , s
2 / n) observation
n
25. Standard Error of the Sample Means
The standard error of the sample mean is
the standard deviation of the sampling
distribution of the sample means.
It is computed by
s = s
n x
x s
is the symbol for the standard error of
the sample mean.
σ is the standard deviation of the
population.
n is the size of the sample.
26. Standard Error of the Sample Means
If s is not known and n ³ 30, the standard
deviation of the sample, designated s , is used
to approximate the population standard
deviation. The formula for the standard error
is:
s s x =
n
27. 95% and 99% Confidence Intervals for
the sample mean
The 95% and 99% confidence intervals are
constructed as follows:
95% CI for the sample mean is given by
m±1.96 s
n
99% CI for the sample mean is given by
m ±2.58 s
n
28. 95% and 99% Confidence Intervals for μ
The 95% and 99% confidence intervals are
constructed as follows:
95% CI for the population mean is given by
X ±1.96 s
n
99% CI for the population mean is given by
X ±2.58 s
n
30. EXAMPLE 3
The Dean of the Business School wants to
estimate the mean number of hours worked
per week by students. A sample of 49
students showed a mean of 24 hours with a
standard deviation of 4 hours. What is the
population mean?
The value of the population mean is not
known. Our best estimate of this value is the
sample mean of 24.0 hours. This value is
called a point estimate.
31. Example 3 continued
Find the 95 percent confidence interval for
the population mean.
1.96 24.00 1.96 4
X s
± = ±
24.00 1.12
49
= ±
n
The confidence limits range from 22.88 to
25.12.
About 95 percent of the similarly constructed
intervals include the population parameter.
32. Confidence Interval for a Population
Proportion
The confidence interval for a population
proportion is estimated by:
p ±z p(1-p)
n
33. EXAMPLE 4
A sample of 500 executives who own their
own home revealed 175 planned to sell their
homes and retire to Arizona. Develop a 98%
confidence interval for the proportion of
executives that plan to sell and move to
Arizona.
.35 ±2.33 (.35)(.65) = ±
.35 .0497
500
34. CONTOH
Lembaga riset melakukan penelitian tentang
perusahaan di Jawa Timur yang sudah
menerapkan UMR. Data menunjukkan dari 50
sampel perusahaan, 40 diantaranya sudah
memenuhi UMR. Buatlah confidence interval
90% untuk menduga persentase perusahaan
yang sudah menerapkan UMR.
35. Finite-Population Correction Factor
A population that has a fixed upper bound is
said to be finite.
For a finite population, where the total
number of objects is N and the size of the
sample is n , the following adjustment is made
to the standard errors of the sample means
and the proportion:
Standard error of the sample means:
N n
s s
= -
-1
N
n x
36. Finite-Population Correction Factor
Standard error of the sample proportions:
N n
p p
= - -
1
(1 )
-
N
n
p s
This adjustment is called the finite-population
correction factor.
If n /N < .05, the finite-population correction
factor is ignored.
37. Finite-Population Correction Factor
Standard error of the sample proportions:
sp
p p
- -
(1 )
n
N n
N
=
-
1
This adjustment is called the finite-population
correction factor.
Note : If n/N < 0.05, the finite-population
correction factor is ignored.
Interval Estimation for proportion with finite-pop
N n
1
ˆ ˆ (1 ˆ )
P p Z p p
= ± - -
-
N
n
38. EXAMPLE 5
Given the information in EXAMPLE 4,
construct a 95% confidence interval for the
mean number of hours worked per week by
the students if there are only 500 students
on campus.
Because n /N = 49/500 = .098 which is
greater than 05, we use the finite
population correction factor.
24 1.96( 4 = ±
) 24.00 1.0648
)( 500 49
± -
500 1
49
-
39. CONTOH
Pimpinan bank ingin mengetahui tentang
kepuasan nasabah terhadap pelayanan bank.
Dari jumlah nasabah 1000 orang, diambil
sampel 100 orang untuk diwawancarai, Hasilnya
60 orang mengakui puas dengan pelayanan
bank tersebut. Dengan a = 5%, berapa proporsi
nasabah yang puas dengan pelayanan bank,
40. Selecting a Sample Size
There are 3 factors that determine the size
of a sample, none of which has any direct
relationship to the size of the population.
They are:
The degree of confidence selected.
The maximum allowable error.
The variation in the population.
41. Selecting a Sample Size
X Z s
n
±( a /2)×s = X ±( Z
a /2)×
X
To find the sample size for a variable:
=æ
z * s
÷ø
çè
ö 2 * = E Þ n E
z s
n
where : E is the allowable error, z is the z-
value corresponding to the selected level of
confidence, and s is the sample deviation of the
pilot survey.
42. EXAMPLE 6
A consumer group would like to estimate the
mean monthly electricity charge for a single
family house in July within $5 using a 99
percent level of confidence. Based on
similar studies the standard deviation is
estimated to be $20.00. How large a sample
is required?
107
(2.58)(20) 2
ö 5
çè
= ÷ø
n = æ
43. Sample Size for Proportions
The formula for determining the sample size in
the case of a proportion is:
2
n = p - p æ
Z
ö çè
÷ø
( 1 ) E
where p is the estimated proportion, based on
past experience or a pilot survey; z is the z
value associated with the degree of confidence
selected; E is the maximum allowable error the
researcher will tolerate.
44. EXAMPLE 7
The American Kennel Club wanted to
estimate the proportion of children that
have a dog as a pet. If the club wanted
the estimate to be within 3% of the
population proportion, how many children
would they need to contact? Assume a
95% level of confidence and that the club
estimated that 30% of the children have a
dog as a pet.
897
(.30)(.70) 1.96
.03
2
= ÷ø
n = æ
ö çè
45. Two-sample Estimation
Mean :
n1, n2 ³ 30
m m X X Z s
1 2 1 2 n
n1, n2 < 30
2
m - m = X - X ± t n - s + n -
s
Proportion :
ö
æ
( ) ÷ ÷ø
ç çè
2
1
- = - ± +
2
2
2
1
s
n
æ
( ) æ
ö
÷ø
÷ + ÷ ÷ø
ç çè
ö
ç çè
( 1) ( 1)
2 2
1 1
n + n -
n n
2
1 2 1 2
1 2 1 2
1 1
2
ˆ ˆ ˆ (1 ˆ ) ˆ (1 ˆ )
P - P = p - p ± Z p - p + -
( )
p p
2 2
2
1 1
1
1 2 1 2
n
n
46. Contoh
Perusahaan ban sedang memebandingkan daya
pakai antara ban merek A dan Merek B. Dari
sampel random 10 ban A diketahui rata-rata
daya pakai 1.000 km dengan standar deviasi
100 km sedangkan dari sampel random 10 ban
merek B rata-rata daya pakai 900 km dengan
standar deviasi 90 km. Hitung perbedaan daya
pakai antara ban merek A dan merek B dengan
a = 0,05.
47. Contoh
Sampel random menunjukkan dari 80 kendaraan
di kota A, 60 diantaranya telah melunasi pajak
sedangkan di kota B dari 70 kendaraan, 40
diantaranya telah melunasi pajak. Hitunglah
perbedaan persentase pelunasan pajak
kendaraan di kedua kota tersebut dengan
tingkat keyakinan 95%.