- The document provides an overview and refresher on key statistical concepts like p-values, alpha levels, z-scores, t-scores, standard errors, and how to perform t-tests. It uses an example of calcium levels in milk bottles to demonstrate these concepts.
- Key points covered include how p-values and alpha levels are used to reject or fail to reject the null hypothesis, how the normal and t-distributions are used, and how standard errors and sample sizes impact results. Examples are worked through to show how to calculate z-scores, t-scores, and interpret the results of t-tests.
Confidence Interval ModuleOne of the key concepts of statist.docxmaxinesmith73660
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval.
The formula that we use to calculate the standard error of the mean is:
s.e. = s / √N – 1
where s = the standard deviation calculated from the sample; and
N = the sample size.
So the formula tells us that the standard error of the mean is equal to the
standard deviation divided by the square root of the sample size minus 1.
This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected.
THE CONFIDENCE INTERVAL (CI)
The formula for the CI is a function of the sample size (N).
For samples sizes ≥ 100, the formula for the CI is:
CI = (the sample mean) + & - Z(s.e.).
Let’s look at an example to see how this formula works.
* Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link.
Example 1
Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
Confidence Interval ModuleOne of the key concepts of statist.docxmaxinesmith73660
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval.
The formula that we use to calculate the standard error of the mean is:
s.e. = s / √N – 1
where s = the standard deviation calculated from the sample; and
N = the sample size.
So the formula tells us that the standard error of the mean is equal to the
standard deviation divided by the square root of the sample size minus 1.
This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected.
THE CONFIDENCE INTERVAL (CI)
The formula for the CI is a function of the sample size (N).
For samples sizes ≥ 100, the formula for the CI is:
CI = (the sample mean) + & - Z(s.e.).
Let’s look at an example to see how this formula works.
* Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link.
Example 1
Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
- Aow
-Aow
f--,
d--Tto o4*prnbAu
SDUJ?_
pya
I
LjrrtripA)c€ IttTS4nOt Mooute
U : YA€aru
! ZG.)=
t,
= rneat-!
Z GltFu)
l
Exa^lrle-4,' N
_
# 44o
bon a*^
wwan=4r.L{'mes
<tabk
S= s.zhnes = 4-96
,,.7,
.XSoo
ll =.Q\So
W=rne&,\s,
Ywe Knot{lWeacL,siob --:4.96s
-
42 ,t
t 4o 5t/. (. arr"[hb,r/olrr.r*WItttt*v
'./\
, sorc/z=.025o=2e/
s5/.C7=
42,4tqlte&r/f,*-ot) to/
35/.eJ=.12.+ g;GZlfi=ee) '
Il :5e/{
gsfT=42,+t4sGGz/fm)
w.Q=,1A4!46(.t+)
gS/U=42+L-53 : 44-tT g5/ cT =
-l
-l-^-
gtJ.CJ=41,4+ .53 = 42.?3 44.{T p42.?3
lt
&.a,Wux: =.4 5oo
67)* , ?w/q
,45oo=z+il)k=Yu(-q*/ r'--=
,O5oit
'f O5oo
@ia/.
eo/8=42.+t4.432/ftu)
1o/.cI=/1.4! tt.o{ztlfrd 3o/.cT=42tt-.qq=
ry
tjc7./2.+t4.658'z/tr(--J31 + -1't=ry
%/,c]=42,q
90/q= 42.1tl,os[,t+1 '/
9r1,cl,424t .rirt
9o/ cf = l,l,g6<+lx,8q
�
eT-flowlb .n1rea/<06!e,rvr-7wz
_
N=flo,(
rneei=/1.4hnes
lt
.i = 3.2 nrn?s
cT_
=42,1J 4as(st/m)
[G)
c n4t L.65(s.21 -{"6ss aq
-421t l-As(z,a/zo) I,6s
= (2 qt 4".65 CIa)
/2'58
t.48 fuo/.c-T=/2-22'+
= /Q.1
:42.4--'rg-'12'2U='12,4+-/[email protected]
=,u<se
Saft&(4OO
t{=47
-Cf
:Wean! +G.")
= tgrFr) .hd00
ftp.a,rLL
oSoo
,/,2,,1
T+o"Ib=Ly
sv/,q=4e.\t1.++6Ss
"/F)
= t1.1! 4.?t^(s./fnt)
= 19..4/ ,t,7//6/3.2/, 9o/ Cf-44,003+>/3.+97
=t27r t.++e70.8)
= ,14,4!4"g97
12.+-/.397=4(.003
4A.4+ /.39/-fi. ?*
�
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the sa ...
WEEK 6 – HOMEWORK 6 LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAP.docxcockekeshia
WEEK 6 – HOMEWORK 6: LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAPTERS 9, 10
INTRODUCTION TO HYPOTHESIS TESTING
WHAT IS A HYPOTHESIS TEST?
Here we are testing claims about the TRUE POPULATION’S STATISTICS based on SAMPLES we have taken. The most common statistic of interest is of course the POPULATION MEAN (µ). But, we can also test its VARIANCE and its STANDARD DEVIATION. (We can also compare TWO or more means to see if there are significant differences.
We must have a basic hypothesis, referred to as the NULL Hypothesis (Ho) and an ALTERNATE Hypothesis (Ha).
Our NULL ( and ALTERNATE) Hypotheses can take three forms:
(1) Ho: µ< some number; Ha: µ > that number (< is “less than or equal to” and > is “greater than or equal to” ),
(2) Ho: µ> some number; Ha: µ < that number , or
(3) Ho: µ = some number; Ha µ≠ that number ; (≠ means “not equal to”)
NOTE THAT Ho MUST HAVE THE “EQUALS” IN IT WHEREAS Ha NEVER DOES.
(1) Is referred to as a “ONE-TAILED TEST TO THE LEFT”
(2) Is a “ONE-TAILED TEST TO THE RIGHT”
(3) Is a “TWO-TAILED TEST”
NEXT, we need to decide what level of significance, i.e.(how sure we want to be about our hypothesis. This is where α comes in again. Do we want to test at the 10%, 5% or 1% level of significance? Another wrinkle is that for the TWO-TAILED test, since our value could be greater OR less than some number, we use α /2 for each extreme, so for 10% it’s 5% (0.050) at each end (tail of the curve), for 5% it’s 2.5% (0.0250) at each end, and for 1% it’s 0.5% (0.0050) at the ends. You have heard about this kind of split before with confidence intervals, but think about it. Here is a graphical display of all this:
As you can see, there is a CRITICAL z-VALUE for each of these test depending on the significance level alpha (α) or α/2.
In HW4 questions 1 and 2, you found the critical z-values for alpha’s of 1%, 5% and 10%, which would work for the one-tailed tests. For the two tailed tests we need to split these alphas (α/2) and find the critical z-values (at the positive and negative tails of the graph) So, for an α of 1% (0.0100) it would be α/2 or 0.005 in the left tail (negative z-value) = -2.575 and for the far right tail (0.005 in that tail) we would have to find the z-value for an area to the LEFT of 99.5% (0.9950) and this is +z = +2.575
Continuing on, for an α of 5% for a two-tailed test the z-values for α/2 would correspond to areas under the curve of 0.0250 at each end. The far left tail would have a negative z-value of -1.96 (see picture above) and the far right tail would have a positive z-value of +1.96 that in the Table represented an area of 97.5% (0.9750) to the LEFT.
Lastly, for an alpha of 10%, hence an α/2 at both ends of 5% (the two-tailed test), the negative z-value would be -1.645.
The positive z-value marking the upper 5% (Table value from 95% to the left) is +1.645.
SO, FOR YOUR USE IN ALL HYPOTHESIS TEST (AND WORKS FOR CONFIDENCE INTERVALS TOO) .
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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.
- Aow
-Aow
f--,
d--Tto o4*prnbAu
SDUJ?_
pya
I
LjrrtripA)c€ IttTS4nOt Mooute
U : YA€aru
! ZG.)=
t,
= rneat-!
Z GltFu)
l
Exa^lrle-4,' N
_
# 44o
bon a*^
wwan=4r.L{'mes
<tabk
S= s.zhnes = 4-96
,,.7,
.XSoo
ll =.Q\So
W=rne&,\s,
Ywe Knot{lWeacL,siob --:4.96s
-
42 ,t
t 4o 5t/. (. arr"[hb,r/olrr.r*WItttt*v
'./\
, sorc/z=.025o=2e/
s5/.C7=
42,4tqlte&r/f,*-ot) to/
35/.eJ=.12.+ g;GZlfi=ee) '
Il :5e/{
gsfT=42,+t4sGGz/fm)
w.Q=,1A4!46(.t+)
gS/U=42+L-53 : 44-tT g5/ cT =
-l
-l-^-
gtJ.CJ=41,4+ .53 = 42.?3 44.{T p42.?3
lt
&.a,Wux: =.4 5oo
67)* , ?w/q
,45oo=z+il)k=Yu(-q*/ r'--=
,O5oit
'f O5oo
@ia/.
eo/8=42.+t4.432/ftu)
1o/.cI=/1.4! tt.o{ztlfrd 3o/.cT=42tt-.qq=
ry
tjc7./2.+t4.658'z/tr(--J31 + -1't=ry
%/,c]=42,q
90/q= 42.1tl,os[,t+1 '/
9r1,cl,424t .rirt
9o/ cf = l,l,g6<+lx,8q
�
eT-flowlb .n1rea/<06!e,rvr-7wz
_
N=flo,(
rneei=/1.4hnes
lt
.i = 3.2 nrn?s
cT_
=42,1J 4as(st/m)
[G)
c n4t L.65(s.21 -{"6ss aq
-421t l-As(z,a/zo) I,6s
= (2 qt 4".65 CIa)
/2'58
t.48 fuo/.c-T=/2-22'+
= /Q.1
:42.4--'rg-'12'2U='12,4+-/[email protected]
=,u<se
Saft&(4OO
t{=47
-Cf
:Wean! +G.")
= tgrFr) .hd00
ftp.a,rLL
oSoo
,/,2,,1
T+o"Ib=Ly
sv/,q=4e.\t1.++6Ss
"/F)
= t1.1! 4.?t^(s./fnt)
= 19..4/ ,t,7//6/3.2/, 9o/ Cf-44,003+>/3.+97
=t27r t.++e70.8)
= ,14,4!4"g97
12.+-/.397=4(.003
4A.4+ /.39/-fi. ?*
�
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the sa ...
WEEK 6 – HOMEWORK 6 LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAP.docxcockekeshia
WEEK 6 – HOMEWORK 6: LANE CHAPTERS, 11, 12, AND 13; ILLOWSKY CHAPTERS 9, 10
INTRODUCTION TO HYPOTHESIS TESTING
WHAT IS A HYPOTHESIS TEST?
Here we are testing claims about the TRUE POPULATION’S STATISTICS based on SAMPLES we have taken. The most common statistic of interest is of course the POPULATION MEAN (µ). But, we can also test its VARIANCE and its STANDARD DEVIATION. (We can also compare TWO or more means to see if there are significant differences.
We must have a basic hypothesis, referred to as the NULL Hypothesis (Ho) and an ALTERNATE Hypothesis (Ha).
Our NULL ( and ALTERNATE) Hypotheses can take three forms:
(1) Ho: µ< some number; Ha: µ > that number (< is “less than or equal to” and > is “greater than or equal to” ),
(2) Ho: µ> some number; Ha: µ < that number , or
(3) Ho: µ = some number; Ha µ≠ that number ; (≠ means “not equal to”)
NOTE THAT Ho MUST HAVE THE “EQUALS” IN IT WHEREAS Ha NEVER DOES.
(1) Is referred to as a “ONE-TAILED TEST TO THE LEFT”
(2) Is a “ONE-TAILED TEST TO THE RIGHT”
(3) Is a “TWO-TAILED TEST”
NEXT, we need to decide what level of significance, i.e.(how sure we want to be about our hypothesis. This is where α comes in again. Do we want to test at the 10%, 5% or 1% level of significance? Another wrinkle is that for the TWO-TAILED test, since our value could be greater OR less than some number, we use α /2 for each extreme, so for 10% it’s 5% (0.050) at each end (tail of the curve), for 5% it’s 2.5% (0.0250) at each end, and for 1% it’s 0.5% (0.0050) at the ends. You have heard about this kind of split before with confidence intervals, but think about it. Here is a graphical display of all this:
As you can see, there is a CRITICAL z-VALUE for each of these test depending on the significance level alpha (α) or α/2.
In HW4 questions 1 and 2, you found the critical z-values for alpha’s of 1%, 5% and 10%, which would work for the one-tailed tests. For the two tailed tests we need to split these alphas (α/2) and find the critical z-values (at the positive and negative tails of the graph) So, for an α of 1% (0.0100) it would be α/2 or 0.005 in the left tail (negative z-value) = -2.575 and for the far right tail (0.005 in that tail) we would have to find the z-value for an area to the LEFT of 99.5% (0.9950) and this is +z = +2.575
Continuing on, for an α of 5% for a two-tailed test the z-values for α/2 would correspond to areas under the curve of 0.0250 at each end. The far left tail would have a negative z-value of -1.96 (see picture above) and the far right tail would have a positive z-value of +1.96 that in the Table represented an area of 97.5% (0.9750) to the LEFT.
Lastly, for an alpha of 10%, hence an α/2 at both ends of 5% (the two-tailed test), the negative z-value would be -1.645.
The positive z-value marking the upper 5% (Table value from 95% to the left) is +1.645.
SO, FOR YOUR USE IN ALL HYPOTHESIS TEST (AND WORKS FOR CONFIDENCE INTERVALS TOO) .
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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.
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
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.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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 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.
For more information, visit-www.vavaclasses.com
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.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2. Refresher
Definition of p-value:
The probability of getting evidence as strong as
you did assuming that the null hypothesis is true.
A smaller p-value means that it’s less likely you would get a
sample like this if the null hypothesis were true.
3. A smaller p-value means stronger evidence against that null
hypothesis.
Definition of alpha, the level of significance:
The highest acceptable p-value that we will use
to reject the null hypothesis.
The default alpha is 0.05.
4. A smaller alpha means less of a chance of falsely rejecting the
null. (Also called a Type I error)
A smaller alpha means we want to be more certain about
something before rejecting the null.
If the p-value is smaller than the alpha, we reject the null
hypothesis. (Enough evidence to reject)
If the p-value is larger than the null, we fail to reject the null
hypothesis. (Not yet enough evidence to reject)
5. Remember this curve? This is the normal curve.
μ, pronounced ‘mu’ is the mean for normals
σ, pronounced ‘sigma’ is the standard deviation for normal
μ + 2σ refers to the point two standard devs above the mean
6. If data follows the normal curve,
about 2/3 (68%) of the data is within 1 standard deviation,
95% of the data is within 2 standard deviations.
7. 2/3 and 95% are proportions, or ratios between a part of a
group and that group as a whole.
Proportions are useful because they also imply probability.
If 2/3 of the data is within 1sd, then if I pick a point at random
from that distribution…
… there is a 2/3 chance that it will be within 1 standard
deviation.
8. Example: Reading scores
Grade 5s reading scores are normally distributed with mean
120 and standard deviation 25.
Pick a grade 5 student at random…
You have a 95% chance of getting one with a reading score
between 70 and 170.
9. Example: Reading scores 2
The normal distribution is symmetric, it’s the same on both
sides of the mean/median.
So the chance of picking a grade 5 with reading score 120 or
more: 0.5
10. Rather than describe things in terms of 'standard deviations
above/below the mean', we shorten this to z-scores.
13. There's also the standard error, which describes the
uncertainty about some measure, such as a sample mean.
14. The standard error is a measure of the typical amount that that
a sample mean will be off from the true mean.
We get the standard error from the standard deviation of the
data…
15. …divided by the square root of the sample size.
- The more variable the distribution is, the more variable
anything based on it, like the mean.
- Standard error increases with standard deviation.
16. - The more we sample from a distribution, the more we know
about its mean.
- Standard error decreases as sample size n increases.
Like standard deviation, the calculation of standard error
isn’t as important as getting a sense of how it behaves.
- Bigger when standard deviation is bigger.
- Smaller when sample size is bigger.
17.
18. Tophat: In order to make a standard error two
times smaller (1/2 as large), what change can
YOU make?
a) Use 1/2 as large a sample.
b) Use 2 times as large a sample.
c) Use 4 times as large a sample.
d) Use 1/2 as large a standard deviation.
19. Tophat: In order to make a standard error two
times smaller (1/5 as large), how many times
as large does the sample need to be?
20. The z-score can be used to describe how many standard
ERRORS above or below a population mean that a sample
mean is. (recall sample mean vs. Population mean)
21.
22. In practice: SPSS and the Milk dataset.
The milk dataset contains a collection of calcium levels of
inspected bottles.
Today we look at a sample of ‘good’ milk where the calcium
level from bottles follows a normal with mean μ = 20, and
standard deviation 5.
From
Analyse Descriptive Statistics Frequencies.
23. …move the variable “Calcium of good sample…” to the right,
and click Statistics.
Select the Mean, Std. Deviation, and S.E. mean, which
stands for Standard Error of mean.
25. The first sample is only of n=4 bottles, so the standard error is
1/ , or ½of the standard deviation.
5.59 / 2 = 2.79
26.
27. Because samples vary, the sample standard deviation, called
s, won’t be exactly the same as the population standard
deviation, σ = 5.
28. Because the standard error of the mean, , is 2.79, it’s not
surprizing that the sample mean is off from the true mean
μ = 20, by 20 – 18.39 = 1.61.
29. Even when μ = 20, we’d expect to see a sample mean of
18.39 or lower…
Z = (18.39 – 20) / (5/ ) = -0.64 TABLE 25.98%
25.98% of the time.
(We use σ = 5 because we were given it in the question)
With a larger sample, we would expect a better estimate.
(This is from the 3rd
sample of 25 bottles)
30. The standard error is smaller (1/5 as much as the std. dev),
And the sample mean is closer to 20.
…and so on. (from the sample of 100 bottles)
31. The sample mean has gotten closer to 20, the sample standard
deviation has gotten closer to 5, and the standard error has
shrank to 1/10 of the standard deviation.
33. … z = -1.01.
So there is a 13.38% chance of getting a sample mean of 20.51
or more when the sample size is 100.
When n=4, the sample mean was off by 1.61, and we found a
25.98% chance of getting a sample mean of that or lower.
When n=100, the sample mean off by 0.50 (three times
closer!), and we found a 15.57% of getting mean of that or
higher.
35. …t-scores.
T-scores are a lot like z-scores, they’re calculated the same way
for sample means.
vs
There is one big difference:
36. Z-scores, based on the normal distribution, use the
population standard deviation.
T-scores, based on the t-distribution, use the sample
standard deviation, s.
37. We use s when we don’t know what the population (true)
standard deviation is.
38. Like the sample mean, which is used when we don’t know the
population mean, the sample standard deviation is used in
place of the population standard deviation when we don’t
know it.
Realistically, μ and σ are parameters, so we never
actually know what they are, so the t-distribution and t-score
are more complicated but more realistic tools than the normal
distribution and z-score.
The t-distribution looks like this…
39. … it’s wider than the normal distribution
The difference gets smaller as we get a larger sample.
40. This is because we get less uncertain about everything,
including the standard deviation as we get a larger sample.
41. If we had no uncertainty about the standard deviation, s would
be the same as sigma, and the t-distribution becomes normal.
42. In fact, the normal and the t-distribution are exactly the same
for infinitely large samples.
44. This is a new sample of milk with population mean 20,
But say we’ve no idea what the true standard deviation is.
We can use the sample standard deviation, s = 5.411, instead.
Now... how likely is it to find a sample with LESS than 18.79
calcium concentration?
First, find the t-score.
46. T= -1.11 Then, use the t-table:
1-sided p=0.25 p=0.10 p=0.05 p=0.025 p=0.010
df
1 1.000 3.078 6.314 12.706 31.821
2 0.816 1.886 2.920 4.303 6.965
3 0.765 1.638 2.353 3.182 4.541
4 0.741 1.533 2.132 2.776 3.747
5 0.727 1.476 2.015 2.571 3.365
… … … … … …
df stands for ‘Degrees of Freedom’. For the t-distribution, this is
n-1. (Sample size minus one).
*** For other analyses, df is found differently, to be covered
later.
47. The sample size n=25, so df = 25 – 1 = 24
Go down the table to df=24, and to the right until you get a
value more than the t-score.
1-sided p=0.25 p=0.10 p=0.05 p=0.025 p=0.010
df
… … … ... … …
22 0.816 1.886 2.920 4.303 6.965
23 0.765 1.638 2.353 3.182 4.541
24 0.741 1.533 2.132 2.776 3.747
25 0.727 1.476 2.015 2.571 3.365
… … … … … …
48. The one sided p values refer to ‘area beyond’ on one side
of the curve.
We can’t get the exact area beyond like we can with a t-score
from this table, but we can get a range.
1-sided p 0.25 0.10
Df
24 0.741 1.533
49. 1-sided p 0.25 0.10
Df
24 0.741 1.533
If 25% of the area of the t24 distribution is higher than 0.741.
And 10% of the area is higher than 1.533.
Then 1.11, which is between 0.741 and 1.533, has an area
beyond somewhere between 10% and 25%.
Without a computer this is the best we can do.
50. 25% of the area is beyond t= -0.741.
There’s a 25% chance of getting a sample mean farther than
0.741 s (sample standard deviations) below the true mean.
51. There’s only a 10% chance of getting a sample mean further
than 1.533 s below the true mean.
52. There is between a 10 and 25% chance of getting a sample
mean further than 1.11 s below the mean.
(By computer the chance is 13.90%)
53. This is starting to sound like a hypothesis test.
With a t-score? Let's call that a....
…t-test.
54. Say we gathered samples from Lake Ontario.
From a sample of 10, the (sample) mean was 23 and the
(sample) standard deviation was 2.5.
If the lake has too much cadmium, we’ll have to take action, so
we want to be sure enough that it’s over the limit so that
there’s only a 0.05 chance of claiming it’s over when it’s not
(i.e. when the null is true)
“The only lake you can develop your film in” – Ron James
55. First, get the t-score
How many standard errors above 20 are we?
56. Get the degrees of freedom, df = n – 1 = 10 – 1= 9.
“ > “, is one sided. (as opposed to “ ”, which is two sided)
For the t-table at df=9 (One sided)
Level of Significance for One-Tailed Test ( )
df .10 .05 .025 .01 .005
… … … … … …
9 1.383 1.833 2.262 2.821 3.250
57. We want to compare our t-score 3.79 to one of these values,
but which one?
We have the magic phrase:
“only a 0.05 chance of claiming it’s over when it’s not”
Which means a level of significance, = .05
Level of Significance for One-Tailed Test ( )
df .10 .05 .025 .01 .005
… … … … … …
9 1.383 1.833 2.262 2.821 3.250
58. For 9 degrees of freedom, there is a .05 chance of getting 1.833
or more standard errors above the mean.
59. There is a less than .05 chance of getting at least 3.75 standard
errors above the mean. (We don’t care how much less)
60. Since getting a t-score of 3.75 is so unlikely, either…
Situation 1:
…the (true) mean concentration of the lake is 20 and we
happened to get a really high concentration sample by dumb
luck, or by chance.
Situation 2:
… the (true) mean concentration is, in fact, higher than 20.
61. The chance of situation 1 happening (high samples by dumb
luck) is less than .05. Since we set the level of significance
is .05, we’re willing to take that chance and...
… reject the null hypothesis.
62. The evidence against the null hypothesis is too strong not to
reject it,
we’re getting samples with concentrations over 20 by a wide
enough margin that it’s not reasonable to say the
concentration is within safe limits.
It’s very unlikely this sample that high by a glitch.
That chance of glitch is called the p-value.
We’ve concluded that the p-value is less than
63. Level of Significance for One-Tailed Test ( )
df .10 .05 .025 .01 .005
… … … … … …
9 1.383 1.833 2.262 2.821 3.250
t-score = 3.75
In fact, the p-value is less than .005, given that t > 3.250.
That’s very strong evidence against the null hypothesis indeed.
64. If the sample mean was fewer standard errors above
20…
65. The p-value could be bigger than = 0.05.
Then there wouldn’t be enough evidence to reject
66. In that case, we fail to reject .
We can’t say that the true mean is exactly 20, because it’s
probably off one way or another. So we don’t accept the null,
we merely fail to reject it.
Even when rejecting , we don’t accept , because
wasn’t the hypothesis being tested.
We statisticians are very sceptical people, never accept, always
reject or fail to reject.
67. Try to grasp the big concepts. They will take care of the little
things.
68.
69. SPSS Example: Milk dataset.
In the milk data set, good milk had a calcium level of 20 mg/L.
Bad milk, for our cases, will have something other than 20.
Step 1: Load the milk dataset.
First we’re going to try goodcalc4, which is a sample of 25
bottles.
70. Our null hypothesis is that the milk is good (calcium is 20mg/L)
The alternative is that the milk has a different calcium value.
71. We’re comparing the mean of one sample to a specific value
(20), so we’ll go to Analyze Compare Means One-Sample
T Test…
72. Choose “ good sample, n=25, 3rd
. Move this variable to the
right by dragging or using the move arrow.
Then, change the test value to 20. (The hypothesized mean)
Click OK
73. You’ll get two tables.
The first is a data summary like what we’ve seen before.
The sample size is 25
The sample mean is 18.79
The sample standard deviation is 5.41115
The standard error of the mean is 1.082
(t-score = -1.12 if we were doing this by hand.)
74. The second table is the result of the t-test
Sig. (2-tailed) is the p-value for a two sided test.
We’re doing a two-sided test (not equal to 20), so this works.
75. p-value = .275 > .050 (default level of significance)
So we fail to reject the null.
Let’s try some bad milk, as found in lowclac2, and this time
let’s say we don’t care if the milk has too much calcium.
76. Another possibility is to include the upper area in the null
hypothesis. The analysis is exactly the same. (Both ways
acceptable)
78. SPSS gives us the two-sided significance, but we only want one
side. By symmetry, we use half of two-sided p-value to get the
one-sided value.
p-value = .016 / 2 = .008 < .05.
We reject the null hypothesis. The milk is bad.
79. Two-tailed means two-sided. The tails are terms for the ends of
a distribution.
Having two tails isn’t normal.
80.
81.
82. Confidence intervals
Say we took a sample of reading speeds of students, and our
sample has a mean of 142 words per minute.
It would be dishonest to say that the population mean reading
speed that uses seatbelts is µ = 142.
But that’s our best guess from our sample, so it would be even
more dishonest to say it’s any other value either.
83. Instead of a single value, it would be a lot more honest to give
an interval that captured most of that uncertainty of the value
and say
“We’re 95% confident that the true parameter (mean,
proportion, whatever) is in this interval.”
The interval we gave would be the 95% confidence interval.
(90%, 99% or other values are possible, but 95% is default, just
like 5% significance.)
84. Subtle note: This doesn’t mean there’s 95% chance that the
true proportion µ is in the interval (.899 to .945). µ is a fixed
value, it’s either in there or it isn’t.
We’ve set an interval that has a 95% chance to contain the
parameter.
85. Each blue vertical line is a confidence interval. The red dotted
line horizontally across them represents the parameter value.
Most blue lines cross the red line (include the parameter), but
not all of them.
86. Confidence interval: Milk example. Find the 95% confidence
interval of the calcium level in the good milk.
F
From SPSS, we know the sample mean is 18.79
and that the standard error of the mean is 1.082
87. The milk example uses a t* critical, because we’re using the t-
distribution.
88. In t-table: One-tailed, .025 significance, df=24, the critical value
is: 2.064. (.050 sig. in two-tailed gives the same value)
We’re 95% confidence that the true calcium level is between...
18.79 – (2.064)(1.082) = 18.79 – 2.23 = 16.56
18.79 + (2.064)(1.082) = 18.79 + 2.23 = 21.02
…is between 16.56 and 21.02.
89. Since the hypothesized value of 20 is within the confidence
interval, 20 is a plausible value for the parameter.
Just as before, we fail to reject the null hypothesis.
We can use the confidence intervals to do two-sided
hypothesis tests as well.
By-hand confidence interval rule:
90. If the confidence interval contains the value
given in the null hypothesis, we fail to reject.
Otherwise, reject.
16.56 to 21.02 contains 20, so we fail to reject
In SPSS, doing a one-sample t-test will automatically give you a
confidence interval as well (defaults to 95%).
91. The values given in SPSS are in relation to 20.
-3.44 means 20 – 3.44 = 16.56
1.02 means 20 + 1.02 = 21.02
SPSS does this because then the reject/fail to reject rule is
simplified a bit:
92. If the confidence interval includes zero, we fail
to reject. Otherwise, reject.
(-3.44 to 1.02 contains zero, so we fail to reject)
93. Final point:
The confidence interval only works for two-sided tests. Why?
The interval cuts off at both ends. It has a lower limit and an
upper limit.
That means if the sample value is too far above or too far
below the null hypothesis value, it will be rejected. By
definition that’s a two-sided test.