Determining the Mean, Variance, and Standard Deviation of a Discrete Random Variable
Visit the website for more services: https://cristinamontenegro92.wixsite.com/onevs
1. Illustrate the t-distribution.
2. Construct the t-distribution.
3. Identify regions under the t-distribution corresponding to different values.
4. Identify percentiles using the t-table.
Visit the website for more services it can offer: https://cristinamontenegro92.wixsite.com/onevs
Probability Distribution (Discrete Random Variable)Cess011697
Learning Competencies:
- to find the possible values of a random variable.
illustrates a probability distribution for a discrete random variable and its properties.
- to compute probabilities corresponding to a given random variable.
There are some exercises for you to answer.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
The steps in computing the median are similar to that of Q1 and Q3
. In finding the median,
we need first to determine the median class. The Q1 class is the class interval where
the 𝑁
4
th score is contained, while the class interval that contains the 3𝑁
4
𝑡ℎ
score is the Q3 class.
Formula :𝑄𝑘 = LB +
𝑘𝑁
4
−𝑐𝑓𝑏
𝑓𝑄𝑘
𝑖
LB = lower boundary of the of the 𝑄𝑘 class
N = total frequency
𝑐𝑓𝑏= cumulative frequency of the class before the 𝑄𝑘 class
𝑓𝑄𝑘
= frequency of the 𝑄𝑘 class
i = size of the class interval
k = the value of quartile being asked
The interquartile range describes the middle 50% of values when
ordered from lowest to highest. To find the interquartile range (IQR),
first find the median (middle value) of the upper and the lower half of
the data. These values are Q1 and Q3
. The IQR is the difference
between Q3 and Q1
.
Interquartile Range (IQR) = Q3 – Q1
The quartile deviation or semi-interquartile range is one-half the
difference between the third and the first quartile.
Quartile Deviation (QD) =
𝑄3−𝑄1
2
The formula in finding the kth decile of a distribution is
𝐷𝑘 = 𝑙𝑏𝑑𝑘 +
(
𝑘
10)𝑁 − 𝑐𝑓
𝑓𝐷𝑘
𝑖
𝐿𝐵𝑑𝑘 − 𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝑁 − 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠
𝑐𝑓 − 𝑐𝑢𝑚𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑏𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝐹𝑑𝑘 − 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝑖 − 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒
As we have learned in the previous lesson, Statistics is a science that studies data. Hence to teach Statistics, real data set is recommend to use. In this lesson, we present an activity where the students will be asked to provide some data that will be submitted for consolidation by the teacher for future lessons. Data on heights and weights, for instance, will be used for calculating Body Mass Index in the integrative lesson. Students will also be given the perspective that the data they provided is part of a bigger group of data as the same data will be asked from much larger groups (the entire class, all Grade 11 students in school, all Grade 11 students in the district). The contextualization of data will also be discussed.
Determining the Mean, Variance, and Standard Deviation of a Discrete Random Variable
Visit the website for more services: https://cristinamontenegro92.wixsite.com/onevs
1. Illustrate the t-distribution.
2. Construct the t-distribution.
3. Identify regions under the t-distribution corresponding to different values.
4. Identify percentiles using the t-table.
Visit the website for more services it can offer: https://cristinamontenegro92.wixsite.com/onevs
Probability Distribution (Discrete Random Variable)Cess011697
Learning Competencies:
- to find the possible values of a random variable.
illustrates a probability distribution for a discrete random variable and its properties.
- to compute probabilities corresponding to a given random variable.
There are some exercises for you to answer.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
The steps in computing the median are similar to that of Q1 and Q3
. In finding the median,
we need first to determine the median class. The Q1 class is the class interval where
the 𝑁
4
th score is contained, while the class interval that contains the 3𝑁
4
𝑡ℎ
score is the Q3 class.
Formula :𝑄𝑘 = LB +
𝑘𝑁
4
−𝑐𝑓𝑏
𝑓𝑄𝑘
𝑖
LB = lower boundary of the of the 𝑄𝑘 class
N = total frequency
𝑐𝑓𝑏= cumulative frequency of the class before the 𝑄𝑘 class
𝑓𝑄𝑘
= frequency of the 𝑄𝑘 class
i = size of the class interval
k = the value of quartile being asked
The interquartile range describes the middle 50% of values when
ordered from lowest to highest. To find the interquartile range (IQR),
first find the median (middle value) of the upper and the lower half of
the data. These values are Q1 and Q3
. The IQR is the difference
between Q3 and Q1
.
Interquartile Range (IQR) = Q3 – Q1
The quartile deviation or semi-interquartile range is one-half the
difference between the third and the first quartile.
Quartile Deviation (QD) =
𝑄3−𝑄1
2
The formula in finding the kth decile of a distribution is
𝐷𝑘 = 𝑙𝑏𝑑𝑘 +
(
𝑘
10)𝑁 − 𝑐𝑓
𝑓𝐷𝑘
𝑖
𝐿𝐵𝑑𝑘 − 𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝑁 − 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠
𝑐𝑓 − 𝑐𝑢𝑚𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑏𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝐹𝑑𝑘 − 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑑𝑒𝑐𝑖𝑙𝑒
𝑖 − 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒
As we have learned in the previous lesson, Statistics is a science that studies data. Hence to teach Statistics, real data set is recommend to use. In this lesson, we present an activity where the students will be asked to provide some data that will be submitted for consolidation by the teacher for future lessons. Data on heights and weights, for instance, will be used for calculating Body Mass Index in the integrative lesson. Students will also be given the perspective that the data they provided is part of a bigger group of data as the same data will be asked from much larger groups (the entire class, all Grade 11 students in school, all Grade 11 students in the district). The contextualization of data will also be discussed.
Lecture 5 Sampling distribution of sample mean.pptxshakirRahman10
Objectives:
Distinguish between the distribution of population and distribution of its sample means
Explain the importance of central limit
theorem
Compute and interpret the standard error of the mean.
Sampling distribution of
sample mean:
A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes.
Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected.
For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population.
The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter. In the above example the average income of households would be the parameter.
The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean.
Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable.
Random variables have distribution ,and since the sample mean is a random variable it must have a distribution.
If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution.
Consider the population with population mean = μ
and standard deviation = σ.
Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means.
This distribution is called the Sampling Distribution of Means.
Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population.
Standard error of the
mean:
The quantity σ is referred to as the standard deviation .it is a measure of spread in the population .
The quality σ/n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean
A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem .
Central Limit Theorem:
Consider a population with finite mean and standard deviation . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples)
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not.
Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study.
The t- and z-test methods developed in the 20th century were used for statistical analysis until 1918, when Ronald Fisher created the analysis of variance method.
ANOVA is also called the Fisher analysis of variance, and it is the extension of the t- and z-tests. The term became well-known in 1925, after appearing in Fisher's book, "Statistical Methods for Research Workers."
It was employed in experimental psychology and later expanded to subjects that were more complex.ANOVA (Analysis Of Variance) is a collection of statistical models used to assess the differences between the means of two independent groups by separating the variability into systematic and random factors. It helps to determine the effect of the independent variable on the dependent variable. Here are the three important ANOVA assumptions:
1. Normally distributed population derives different group samples.
2. The sample or distribution has a homogenous variance
3. Analysts draw all the data in a sample independently.
ANOVA test has other secondary assumptions as well, they are:
1. The observations must be independent of each other and randomly sampled.
2. There are additive effects for the factors.
3. The sample size must always be greater than 10.
4. The sample population must be uni-modal as well as symmetrical.
TYPES OF ANOVA
1. One way ANOVA analysis of variance is commonly called a one-factor test in relation to the dependent subject and independent variable. Statisticians utilize it while comparing the means of groups independent of each other using the Analysis of Variance coefficient formula. A single independent variable with at least two levels. The one way Analysis of Variance is quite similar to the t-test.
2 TWO WAY ANOVA
The pre-requisite for conducting a two-way anova test is the presence of two independent variables; one can perform it in two ways –
Two way ANOVA with replication or repeated measures analysis of variance – is done when the two independent groups with dependent variables do different tasks.
Two way ANOVA sans replication – is done when one has a single group that they have to double test like one tests a player before and after a football game
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
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.
3. Mean & Variance of
Sampling
Distributions of
Sample Means
A Grade 11 Statistics and Probability Lecture
4. At the end of this lesson, you are
expected to:
find the mean and variance of
the sampling distribution of the
sample means.
state and explain the Central
Limit Theorem.
use the Central Limit Theorem to
solve problems involving means
and variances of sampling
distribution of means.
Lesson Objectives
Mean and Variance of Sampling Distributions of Sample Means
5. Review: Finding the Mean and Variance of Discrete
Probability Distributions
CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
X P(X)
1 0.1
2 0.3
3 0.2
4 0.4
X P(X)
0.1
0.6
0.6
1.6
X2P(X)
0.1
1.2
1.8
6.4
2.9 9.5SUMS
Mean: 2.9
Variance:
2 2
9.5 2.9
1.09
Standard deviation:
1.09 1.04
6. Recall:
Sampling Distribution of the Mean
CABT Statistics & Probability – Grade 11 Lecture Presentation
A sampling distribution of
sample means is a frequency
distribution using the means
computed from all possible
random samples of a specific
size taken from a population.
The probability distribution of the sample
means is also called the sampling distribution
of the sample means, with the sample mean as
the random variable.
http://www.philender.com/courses/intro/notes2/sam.gif
Mean and Variance of Sampling Distributions of Sample Means
7. Recall: Constructing Sampling Distribution
of Sample Means
CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
How to Construct a Sampling
Distribution of Sample Means from a
Given PopulationSTEP 1 – Determine the number of
samples of size n from the population of
size N.
STEP 2 – List all the possible samples
and compute the mean of each sample.
STEP 3 – Construct a frequency
distribution of the sample means
obtained in Step 2.
STEP 4 – Construct the probability
8. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling
Distribution of Sample Means
Because the sampling distribution P(X)
of
sample means X is essentially a
discrete probability distribution, the
computation of the mean and variance
of P(X) is the SAME as for any discrete
probability distribution P(X).
Ows,
talaga?
Mismo!
9. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling
Distribution of Sample Means
For the sampling distribution of
the sample means :
Mean: X
X P X g
Varianc
e:
2 2 2
X X
X P X g
Standard
deviation:
2
X X
P X
X
10. CABT Statistics & Probability – Grade 11 Lecture Presentation
Consider a group of N = 4 people
with the following ages: 18, 20, 22,
24. Consider samples of size n = 2
from the group without
replacement.
If X is the age of one person in
the group and X is the average
age of the two people in a
sample, find the mean and
variance of the sampling
distribution of X.
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
11. CABT Statistics & Probability – Grade 11 Lecture Presentation
Sampling without replacement
Number of samples of size 2:4 2
6C
Sample
Sample
Mean
18, 20 19
18, 22 20
18, 24 21
20, 22 21
20, 24 22
22, 24 23
Sample
Mean
Frequency Probability
19 1 1/6
20 1 1/6
21 2 1/3
22 1 1/6
23 1 1/6
TOTAL 6 1
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
12. CABT Statistics & Probability – Grade 11 Lecture Presentation
Sample
Mean
Frequency Probability
19 1 1/6
20 1 1/6
21 2 1/3
22 1 1/6
23 1 1/6
TOTAL 6 1
X P(X)
19 1/6
20 1/6
21 1/3
22 1/6
23 1/6
Mean and Variance of Sampling Distributions of Sample Means
Sampling without replacement
Number of samples of size 2:4 2
6C
The Mean and Variance of Sampling Distribution
of Sample Means
13. CABT Statistics & Probability – Grade 11 Lecture Presentation
Computing the mean and
variance:
Mean and Variance of Sampling Distributions of Sample Means
X P(X)
19 1/6
20 1/6
21 1/3
22 1/6
23 1/6
X P(X) X2 P(X)
3.17 60.17
3.33 66.67
7.00 147.00
3.67 80.67
3.83 88.17
21.00 442.67SUMS
Mean: 21X
Variance:
2 2
442.67 21X
1.67
The Mean and Variance of Sampling Distribution
of Sample Means
14. CABT Statistics & Probability – Grade 11 Lecture Presentation
Redo Example 1 if sampling is with
replacement.
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling
Distribution of Sample Means
Sampling with replacement
Number of samples of size 2: 2
4 16
Sample Mean Sample Mean Sample Mean Sample Mean
18,18 18 20, 18 19 22,18 20 24,18 21
18, 20 19 20, 20 20 22, 20 21 24, 20 22
18, 22 20 20, 22 21 22, 22 22 24, 22 23
18, 24 21 20, 24 22 22, 24 23 24, 24 24
15. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
Sampling with replacement
Number of samples of size 2: 2
4 16
Sample
Mean
Frequency Probability
18 1 1/16
19 2 1/8
20 3 3/16
21 4 1/4
22 3 3/16
23 2 1/8
24 1 1/16
TOTAL 16 1
X P(X)
18 1/16
19 1/8
20 3/16
21 1/4
22 3/16
23 1/8
24 1/16
The Mean and Variance of Sampling Distribution
of Sample Means
16. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling
Distribution of Sample Means
X P(X)
18 1/16
19 1/8
20 3/16
21 1/4
22 3/16
23 1/8
24 1/16
Computing the mean and variance:
X P(X) X2 P(X)
1.13 20.25
2.38 45.13
3.75 75.00
5.25 110.25
4.13 90.75
2.88 66.13
1.50 36.00
21 443.5SUMS
Mean: 21X
Variance:
2 2
443.5 21X
2.5
17. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
Consider a population with values
1, 1, 2, 3. Samples of size 3 are drawn
from the population without
replacement.
a. Construct a sampling distribution
of the sample means for this
population.
b. Find the mean, variance, and
standard deviation of the
sampling distribution.SOLUTION HERE!
The Mean and Variance of Sampling Distribution
of Sample Means
20. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
QUESTION!
How are the ACTUAL mean and
variance of the population
related to the mean and
variance of the sampling
distribution?
The Mean and Variance of Sampling Distribution
of Sample Means
21. CABT Statistics & Probability – Grade 11 Lecture Presentation
Example 1 and 2 revisited:
Population: (18, 20, 22, 24)
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
Mean of the population:
18 20 22 24
21
4
X X - (X - )2
18 3 9
20 1 1
22 1 1
24 3 9
sum 20
Variance of the population:
2
2
X
N
20
5
4
22. CABT Statistics & Probability – Grade 11 Lecture Presentation
Let’s compare!
Mean and Variance of Sampling Distributions of Sample Means
Mean Variance
Population
Sampling
Distribution
(samples of size 2 without replacement)
21
21X
2
5
2
1.67X
Population: (18, 20, 22, 24)
Sampling: n = 2, without replacement
The Mean and Variance of Sampling Distribution
of Sample Means
23. CABT Statistics & Probability – Grade 11 Lecture Presentation
Let’s compare!
Mean and Variance of Sampling Distributions of Sample Means
Mean Variance
Population
Sampling
Distribution
(samples of size 2 with replacement)
21
21X
2
5
2
2.5X
Population: (18, 20, 22, 24)
Sampling: n = 2, with replacement
The Mean and Variance of Sampling Distribution
of Sample Means
24. CABT Statistics & Probability – Grade 11 Lecture Presentation
Let’s compare!
Mean and Variance of Sampling Distributions of Sample Means
Values of variances (N = 4, n = 2)
The Mean and Variance of Sampling Distribution
of Sample Means
Without replacement:
2
2 5 4 2
1.67
2 4 1 1X
N n
n N
With replacement:
2
2 5
2.5
2X
n
25. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
Properties of the Mean and Variance of Sampling
Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
Condition MEAN VARIANCE
For infinite
populations or for
sampling with
replacement
For finite
populations or for
sampling without
replacement
X
X
2
2
X
n
2
2
1X
N n
n N
26. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
Properties of the Mean and Variance of
Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
1. The mean of the sample means will
be the same as the population mean.
2. The variance or standard deviation of
the sample means will be smaller
than the variance or standard
deviation of the population.
27. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
If a population is normal with mean μ and
standard deviation σ, the sampling distribution
of
is also normally distributed with the
following:
X
The Mean and Variance of Sampling Distribution
of Sample Means
mean standard deviation*variance
X
2
2
X
n
X
n
*Note: For , you can also useX
2
X X
28. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
If the population is infinite with mean μ and
standard deviation σ or the sampling is with
replacement, use the following:
mean standard deviation*variance
X
2
2
X
n
X
n
The Mean and Variance of Sampling Distribution
of Sample Means
*Note: For , you can also useX
2
X X
29. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
If the population is finite with mean μ and
standard deviation σ or the sampling is
without replacement, use the following:
mean standard deviation*variance
X
2
2
1X
N n
n N
The Mean and Variance of Sampling Distribution
of Sample Means
1X
N n
Nn
*Note: For , you can also useX
2
X X
31. For sampling with replacement:
Larger
sample size
Smaller
sample size
xμ
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
As the sample size n increases,
decreases.
X
32. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Standard Error of the Mean
A measure of the variability in the mean from
sample to sample is given by the Standard Error
of the Mean, which is simply equal to the standard
deviation of the sampling distribution of the sample
means :
Note that the standard error of the mean
decreases as the sample size increases.
X
X
n
1X
N n
Nn
33. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Finite Population
Correction Factor
1
N n
N
The FINITE POPULATION CORRECTION
FACTOR is used for computing the standard
error of the mean from a finite population. It is
given by the expression
Wow!
34. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Finite Population
Correction Factor
The correction factor is necessary if
relatively large samples are taken from a
small population, because the sample mean
will then more accurately estimate the
population mean and there will be less error
in the estimation.
Why the need for the
correction factor? 1
N n
N
35. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Finite Population
Correction Factor
The finite population correction factor is
also used when the sample size n is
LARGE relative to the population N.
This means that n is larger than 5% of
N:
0.05n N
For example, if N = 10 and n = 6,
6 0.05 10 5
so the correction factor is needed in this case.
36. CABT Statistics & Probability – Grade 11 Lecture Presentation
Consider an infinite population
with mean 20 and standard
deviation 2. Samples of size 4 are
obtained from the population.
Determine the following:
a. Mean of the sampling
distribution
b. Variance of the standard
distribution
c. Standard error of the mean
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
37. Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distribution
of Sample Means
CABT Statistics & Probability – Grade 11 Lecture Presentation
Given: = 20, = 2, n = 4
Since the population is normal and
infinite, we use the following formulas:
mean
sampling error of
the mean
(standard deviation)
variance
X
2
2
X
n
X
n
20
2
2
4
1
2
4
1
38. CABT Statistics & Probability – Grade 11 Lecture Presentation
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling
Distribution of Sample Means
Consider a finite population with
size 10 with mean 20 and standard
deviation 2. Samples of size 4 are
obtained from the population
without replacement. Determine the
following:
a. Mean of the sampling distribution
b. Variance of the standard
distribution
c. Standard error of the mean
39. Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distributions
of Sample Means
CABT Statistics & Probability – Grade 11 Lecture Presentation
Given: = 20, = 2, n = 4, N = 10
Since the population is finite and sampling is
without replacement, we use the following
formulas:
mean
sampling error of
the mean
(standard deviation)
variance
X
2
2
1X
N n
n N
2
X X
20
2
2 10 4
4 10 1
2
0.67X
0.67 0.82
40.
41. CABT Statistics & Probability – Grade 11 Lecture Presentation
If a population is normal with mean μ and standard
deviation σ, the sampling distribution of the sample
means is also normally distributed, regardless of the
sample size.
What can we say about the shape of the
sampling distribution of the sample means
when the population from which the sample is
selected is not normal?
QUESTION!
Mean and Variance of Sampling Distributions of Sample Means
The Mean and Variance of Sampling Distributions
of Sample Means
43. CABT Statistics & Probability – Grade 11 Lecture Presentation
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.
Mean and Variance of Sampling Distributions of Sample Means
The Central Limit Theorem
44. CABT Statistics & Probability – Grade 11 Lecture Presentation
What does this
theorem mean?
If the sample size n drawn from a finite
population of any shape is LARGE enough,
then it is safe to ASSUME that the distribution
is APPROXIMATELY NORMAL, and thus the
following can be used:
X
X
n
Mean and Variance of Sampling Distributions of Sample Means
The Central Limit Theorem
45. CABT Statistics & Probability – Grade 11 Lecture Presentation
How largeis
“largeenough”?
According to most statistics books,
the sample size n is LARGE
ENOUGH when…
30n
Wow!
Mean and Variance of Sampling Distributions of Sample Means
The Central Limit Theorem
47. The Central
Limit Theorem
ILLUSTRATION
OF THE CENTRAL
LIMIT THEOREM
FOR THREE
POPULATIONS
(from Statistics for Business and
Economics by Anderson, Sweeney and
Williams, 10th ed)
48. CABT Statistics & Probability – Grade 11 Lecture Presentation
Two things to remember in using the Central Limit
Theorem:
1. When the original variable is normally
distributed, the distribution of the sample means
will be normally distributed for any sample size
n.
2. When the distribution of the original variable
might not be normal, a sample size of 30 or
more is needed to use a normal distribution to
approximate the distribution of the sample
means. The larger the sample, the better the
approximation will be.
Mean and Variance of Sampling Distributions of Sample Means
The Central Limit Theorem
49. CABT Statistics & Probability – Grade 11 Lecture Presentation
RECALL: If the distribution of a random
variable X is normal with mean and standard
deviation , the equivalent z-scores is obtained
by using
x
z
This formula transforms the values of the
variable x into standard units or z values.
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
50. CABT Statistics & Probability – Grade 11 Lecture Presentation
For a normally (or approximately normally)
distributed population with mean and
standard deviation , the equivalent z-score of
a sample mean from the sampling
distribution of means is given by the formula
x
x
x
z
x
z
n
or
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
51. CABT Statistics & Probability – Grade 11 Lecture Presentation
Distribution of Sample Means for a Large Number of
Samples
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
52. CABT Statistics & Probability – Grade 11 Lecture Presentation
A population is normally distributed
with mean 20 and standard deviation
6. Samples of size 9 are drawn from
the population. What are the
equivalent z-scores of the following
sample means?a. b.22x 25x
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
53. CABT Statistics & Probability – Grade 11 Lecture Presentation
A population is normally
distributed with mean 20 and
standard deviation 6. Samples of
size 9 are drawn from the
population. What is the
probability that the sample mean
is
a. greater than 22?
b. between 22 and 25?
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
54. CABT Statistics & Probability – Grade 11 Lecture Presentation
A population has mean 50 and
standard deviation 12. Samples
of size 36 are drawn from the
population. What is the
probability that the sample mean
is
a. less than 48?
b. between 42 and 50?
c. greater than 53.5?
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
55. CABT Statistics & Probability – Grade 11 Lecture Presentation
A recent study of a company regarding the
lifetimes of cell phones it issues to its
employees revealed that the average is
24.3 months and the standard deviation is
2.6 months.
a. If a company phone is chosen at
random, what is the probability that the
phone will last for more than 3 years?
b. If a company randomly samples 40
phones, what is the probability that the
average lifetime of the sample is more
than 3 years?
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
56. CABT Statistics & Probability – Grade 11 Lecture Presentation
Exercise 4 on page 133, edited
In a study of life expectancy of 400 people
in a certain geographic region, the mean
age of death is 70 years and the standard
deviation is 5.1 years. If a sample of 50
people from this region is selected, what
is the probability that the mean life
expectancy is less than 68 years?
Mean and Variance of Sampling Distributions of Sample Means
Application of the
Central Limit Theorem
57. CABT Statistics & Probability – Grade 11 Lecture Presentation
Check your
understandingDo Exercise
2 on page
132 and
Extension
A#1 on p.
133 of your
Mean and Variance of Sampling Distributions of Sample Means
58. CABT Statistics & Probability – Grade 11 Lecture Presentation
Assignment
Do Exercise #1 on
p. 132, due on
Thursday,
August 25.
Mean and Variance of Sampling Distributions of Sample Means
Quiz on the Central
Limit Theorem on
Tuesday, September
62. CABT Statistics & Probability – Grade 11 Lecture Presentation
Quiz #1
A population has a mean of 90 and
standard deviation 3.5. Samples of size
49 are taken from the population and the
sample means are taken.
1. Determine the z-scores of the
following sample means:
Mean and Variance of Sampling Distributions of Sample Means
88.5 90.15a x b x
2. What is the probability that the
sample mean will lie between 88.5 and
90.15?