NUMERICA METHODS 1 final touch summary for test 1musadoto
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
NUMERICA METHODS 1 final touch summary for test 1musadoto
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
Statistical inference: Probability and DistributionEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 1 (probability) and week 2 (distribution).
Descriptive Statistics Formula Sheet Sample Populatio.docxsimonithomas47935
Descriptive Statistics Formula Sheet
Sample Population
Characteristic statistic Parameter
raw scores x, y, . . . . . X, Y, . . . . .
mean (central tendency) M =
∑ x
n
μ =
∑ X
N
range (interval/ratio data) highest minus lowest value highest minus lowest value
deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )
average deviation (average
distance from mean)
∑(x − M )
n
= 0
∑(X − μ )
N
sum of the squares (SS)
(computational formula) SS = ∑ x
2 −
(∑ x)2
n
SS = ∑ X2 −
(∑ X)2
N
variance ( average deviation2 or
standard deviation
2
)
(computational formula)
s2 =
∑ x2 −
(∑ x)2
n
n − 1
=
SS
df
σ2 =
∑ X2 −
(∑ X)2
N
N
standard deviation (average
deviation or distance from mean)
(computational formula) s =
√∑ x
2 −
(∑ x)2
n
n − 1
σ =
√∑ X
2 −
(∑ X)2
N
N
Z scores (standard scores)
mean = 0
standard deviation = ± 1.0
Z =
x − M
s
=
deviation
stand. dev.
X = M + Zs
Z =
X − μ
σ
X = μ + Zσ
Area Under the Normal Curve -1s to +1s = 68.3%
-2s to +2s = 95.4%
-3s to +3s = 99.7%
Using Z Score Table for Normal Distribution
(Note: see graph and table in A-23)
for percentiles (proportion or %) below X
for positive Z scores – use body column
for negative Z scores – use tail column
for proportions or percentage above X
for positive Z scores – use tail column
for negative Z scores – use body column
to discover percentage / proportion between two X values
1. Convert each X to Z score
2. Find appropriate area (body or tail) for each Z score
3. Subtract or add areas as appropriate
4. Change area to % (area × 100 = %)
Regression lines
(central tendency line for all
points; used for predictions
only) formula uses raw
scores
b = slope
a = y-intercept
y = bx + a
(plug in x
to predict y)
b =
∑ xy −
(∑ x)(∑ y)
n
∑ x2 −
(∑ x)2
n
a = My - bMx
where My is mean of y
and Mx is mean of x
SEest (measures accuracy of predictions; same properties as standard deviation)
Pearson Correlation Coefficient
(used to measure relationship;
uses Z scores)
r =
∑ xy−
(∑ x)(∑ y)
n
√(∑ x2−
(∑ x)2
n
)(∑ y2−
(∑ y)2
n
)
r =
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦
r
2
= estimate or % of accuracy of predictions
PSYC 2317 Mark W. Tengler, M.S.
Assignment #9
Hypothesis Testing
9.1 Briefly explain in your own words the advantage of using an alpha level (α) = .01
versus an α = .05. In general, what is the disadvantage of using a smaller alpha
level?
9.2 Discuss in your own words the errors that can be made in hypothesis testing.
a. What is a type I error? Why might it occur?
b. What is a type II error? How does it happen?
9.3 The term error is used in two different ways in the context of a hypothesis test.
First, there is the concept of sta
Statistical inference: Probability and DistributionEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 1 (probability) and week 2 (distribution).
Descriptive Statistics Formula Sheet Sample Populatio.docxsimonithomas47935
Descriptive Statistics Formula Sheet
Sample Population
Characteristic statistic Parameter
raw scores x, y, . . . . . X, Y, . . . . .
mean (central tendency) M =
∑ x
n
μ =
∑ X
N
range (interval/ratio data) highest minus lowest value highest minus lowest value
deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )
average deviation (average
distance from mean)
∑(x − M )
n
= 0
∑(X − μ )
N
sum of the squares (SS)
(computational formula) SS = ∑ x
2 −
(∑ x)2
n
SS = ∑ X2 −
(∑ X)2
N
variance ( average deviation2 or
standard deviation
2
)
(computational formula)
s2 =
∑ x2 −
(∑ x)2
n
n − 1
=
SS
df
σ2 =
∑ X2 −
(∑ X)2
N
N
standard deviation (average
deviation or distance from mean)
(computational formula) s =
√∑ x
2 −
(∑ x)2
n
n − 1
σ =
√∑ X
2 −
(∑ X)2
N
N
Z scores (standard scores)
mean = 0
standard deviation = ± 1.0
Z =
x − M
s
=
deviation
stand. dev.
X = M + Zs
Z =
X − μ
σ
X = μ + Zσ
Area Under the Normal Curve -1s to +1s = 68.3%
-2s to +2s = 95.4%
-3s to +3s = 99.7%
Using Z Score Table for Normal Distribution
(Note: see graph and table in A-23)
for percentiles (proportion or %) below X
for positive Z scores – use body column
for negative Z scores – use tail column
for proportions or percentage above X
for positive Z scores – use tail column
for negative Z scores – use body column
to discover percentage / proportion between two X values
1. Convert each X to Z score
2. Find appropriate area (body or tail) for each Z score
3. Subtract or add areas as appropriate
4. Change area to % (area × 100 = %)
Regression lines
(central tendency line for all
points; used for predictions
only) formula uses raw
scores
b = slope
a = y-intercept
y = bx + a
(plug in x
to predict y)
b =
∑ xy −
(∑ x)(∑ y)
n
∑ x2 −
(∑ x)2
n
a = My - bMx
where My is mean of y
and Mx is mean of x
SEest (measures accuracy of predictions; same properties as standard deviation)
Pearson Correlation Coefficient
(used to measure relationship;
uses Z scores)
r =
∑ xy−
(∑ x)(∑ y)
n
√(∑ x2−
(∑ x)2
n
)(∑ y2−
(∑ y)2
n
)
r =
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦
r
2
= estimate or % of accuracy of predictions
PSYC 2317 Mark W. Tengler, M.S.
Assignment #9
Hypothesis Testing
9.1 Briefly explain in your own words the advantage of using an alpha level (α) = .01
versus an α = .05. In general, what is the disadvantage of using a smaller alpha
level?
9.2 Discuss in your own words the errors that can be made in hypothesis testing.
a. What is a type I error? Why might it occur?
b. What is a type II error? How does it happen?
9.3 The term error is used in two different ways in the context of a hypothesis test.
First, there is the concept of sta
This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxanhlodge
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 bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the result
This is t.
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxagnesdcarey33086
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 bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the result
This is t.
SAMPLING MEAN DEFINITION The term sampling mean is.docxagnesdcarey33086
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical
distributions. In statistical terms, the sample mean from 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 bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, µ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
µ.
1. Rentang
2. Rentang antarkuartil
3. Rentang semikuartil
4. Rentang a-b persentil
5. Simpangan baku
6. Variansi
7. Ukuran penyebaaran relatif
8. Bilangan baku
[The following information applies to the questions displayed belo.docxdanielfoster65629
[The following information applies to the questions displayed below.]
A sample of 36 observations is selected from a normal population. The sample mean is 12, and the population standard deviation is 3. Conduct the following test of hypothesis using the 0.01 significance level.
H0: μ ≤ 10
H1: μ > 10
1.
Value:
10.00 points
Required information
a.
Is this a one- or two-tailed test?
One-tailed test
Two-tailed test
References
EBook & Resources
Multiple Choice Difficulty: 2 Intermediate Learning Objective: 10-05 Conduct a test of a hypothesis about a population mean.
eBook: Conduct a test of a hypothesis about a population mean.
Check my work
2.
Value:
10.00 points
Required information
b.
What is the decision rule?
Reject H0 when z ≤ 2.326
Reject H0 when z > 2.326
References
EBook & Resources
Multiple Choice Difficulty: 2 Intermediate Learning Objective: 10-05 Conduct a test of a hypothesis about a population mean.
eBook: Conduct a test of a hypothesis about a population mean.
Check my work
3.
Value:
10.00 points
Required information
c.
What is the value of the test statistic?
Value of the test statistic
References
EBook & Resources
Worksheet Difficulty: 2 Intermediate Learning Objective: 10-05 Conduct a test of a hypothesis about a population mean.
eBook: Conduct a test of a hypothesis about a population mean.
Check my work
4.
Value:
10.00 points
Required information
d.
What is your decision regarding H0?
Fail to reject H0
Reject H0
References
EBook & Resources
Multiple Choice Difficulty: 2 Intermediate Learning Objective: 10-05 Conduct a test of a hypothesis about a population mean.
eBook: Conduct a test of a hypothesis about a population mean.
Check my work
5.
Value:
10.00 points
Required information
e.
What is the p-value?
p-value
References
Given the following hypotheses:
H0 : μ = 400
H1 : μ ≠ 400
A random sample of 12 observations is selected from a normal population. The sample mean was 407 and the sample standard deviation 6. Using the .01 significance level:
a.
State the decision rule. (Negative amount should be indicated by a minus sign. Round your answers to 3 decimal places.)
Reject H0 when the test statistic is the interval (,).
b.
Compute the value of the test statistic. (Round your answer to 3 decimal places.)
Value of the test statistic
c.
What is your decision regarding the null hypothesis?
Do not reject
Reject
The management of White Industries is considering a new method of assembling its golf cart. The present method requires 42.3 minutes, on the average, to assemble a cart. The mean assembly time for a random sample of 24 carts, using the new method, was 40.6 minutes, and the standard deviation of the sample was 2.7 minutes. Using the .10 level of significance, can we conclude that the assembly time using the new method is faster?
a.
What is the decision rule? (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)
Rej.
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C2 st lecture 10 basic statistics and the z test handout
1. Lecture 10 - Basic Statistics and the Z-test
C2 Foundation Mathematics (Standard Track)
Dr Linda Stringer Dr Simon Craik
l.stringer@uea.ac.uk s.craik@uea.ac.uk
INTO City/UEA London
2. Lecture 9 skills
Calculate the following measures of location (AVERAGES)
Mode
Median
Mean
Calculate the following measures of dispersion
(MEASURES OF SPREAD)
Interquartile range
Standard deviation
Absolute deviation
Perform a Z-test
Write the null and alternative hypothesis
Look up the critical value
Calculate the test statistic
Make the decision
Write a conclusion
3. A data set
A data set is usually a list of values (numbers) that has
been gathered in a survey.
We will use the following data set to demonstrate the ideas
in the first part of this lecture.
A statistician wants to find how many pets the average
person has. He interviews 10 people and gets the following
values
0 2 0 1 0 8 2 1 0 0
4. Bar charts
A bar chart showing how many pets 10 people have:
0 2 0 1 0 8 2 1 0 0
1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
5. Pie charts
A pie chart of the data
0 2 0 1 0 8 2 1 0 0
0
50%
1
20%
2
20% 8
10%
6. Histogram
A histogram of the data showing how many people have each
number of pets.
0 2 0 1 0 8 2 1 0 0
0 1 2 8
1
2
3
4
5
7. Mode
In a data set the mode is the most frequent value (the value
which occurs most often). The mode is a type of average.
Example: Find the mode of the following data set
0 2 0 1 0 8 2 1 0 0
In this data set the mode is 0.
8. Mode
There can be more than one mode in a data set
Example:
0 5 5 0 1 5 0 1 6
There are two modes, they are 0 and 5.
9. Median
The median is the middle value in an ordered data set. It is
another type of average.
First order the data, with values increasing from left to right.
Let n be the size of the data set (the number of values).
If n
2 is an integer (whole number) then the median is the
midpoint of the n
2 th value and the n
2 + 1th value (to find the
midpoint, add the values together and divide by 2).
If n
2 is not an integer (whole number) then round it up to the
nearest integer (n+1
2 ). The median is the n+1
2 th value.
OR find the median by crossing off pairs of values, starting
from the ends of the data set.
10. Example
Order the data:
0 0 0 0 0 1 1 2 2 8
n = 10 (the number of values)
n
2 = 10
2 = 5, which is an integer
The median is the midpoint of the 5th and 6th value =
0+1
2 = 0.5.
11. Example 2
Order the data:
0 0 0 1 1 5 5 5 6
n = 9 (the number of values)
n
2 = 9
2 = 4.5, which is not an integer.
Round up to 5. The median is the 5th value = 1.
12. Interquartile range
First order the data, with values increasing from left to right.
We want to find two values: the first quartile Q1 and the
third quartile Q3.
Let n be the size of the data set (the number of values).
To find Q1 we multiply n by 1
4 . If n
4 is an integer (whole
number) then Q1 is the midpoint of the (n
4 )th value and the
(n
4 + 1)th value
If n
4 is not an integer then round it up to the nearest integer.
Q1 is the corresponding value.
To find Q3 we multiply n by 3
4 . If 3n
4 is an integer then Q3 is
the midpoint of the (3n
4 )th value and the (3n
4 + 1)th value
If 3n
4 is not an integer then round it up to the nearest
integer. Q3 is the corresponding value.
The interquartile range is Q3 − Q1.
13. Example
Order the data
0 0 0 0 0 1 1 2 2 8
n
4 = 10
4 = 2.5, which is not an integer.
Round up to 3.
Q1 is the third value, so Q1 = 0.
3n
4 = 3×10
4 = 7.5, which is not an integer.
Round up to 8.
Q3 is the eighth value, so Q3 = 2.
The interquartile range is Q3 − Q1 = 2 − 0 = 2.
14. Sigma notation Σ
Given a data set X, we denote the sum of all the values x
in X by
x
Example: If
X = 0 2 0 1 0 8 2 1 0 0
then x = 0 + 2 + 0 + 1 + 0 + 8 + 2 + 1 + 0 + 0 = 14
15. Mean
The mean is our third average.
In a data set of size n the mean, denoted ¯x, is the sum of
all the values divided by n.
¯x =
x
n
Example: What is the mean number of pets?
Calculate the sum of all the values and divide by n
¯x =
x
n
=
0 + 2 + 0 + 1 + 0 + 8 + 2 + 1 + 0 + 0
10
=
14
10
= 1.4
16. Standard deviation, σ
The standard deviation, σ is a measure of dispersion.
First calculate the variance, σ2. The standard deviation, σ,
is the square root of the variance.
There are two formulae for variance. They give the same
answer. Usually the second formula is easier to use.
σ2
=
(x − ¯x)2
n
=
x2
n
− ¯x2
When you have found the variance, do not forget to take
the square root !
σ =
x2
n
− ¯x2
17. Proof that the two formulae for standard deviation are
equivalent
σ2
= (x−¯x)2
n
= x2
−2x¯x+¯x2
n
= x2
n − 2¯x x
n +
¯x2
n
= x2
n − 2¯x2
+ ¯x2 1
n
= x2
n − ¯x2
18. Example
What is the standard deviation of the following data ?
0 2 0 1 0 8 2 1 0 0
Use the second formula to calculate the variance.
σ2
=
x2
n
− ¯x2
We previously worked out the mean ¯x = 1.4.
x2
= 02
+22
+02
+12
+02
+82
+22
+12
+02
+02
= 74
The variance is
σ2
=
x2
n
− ¯x2
=
74
10
− 1.42
= 5.44
The standard deviation is σ =
√
5.44 = 2.33 to 2 d.p.
19. Absolute value
The absolute value function gives the positive value of any
number
|x| =
x if x ≥ 0
−x if x < 0
|5| = 5,
| − 8| = 8,
| − 1.213| = 1.213.
|1, 000, 000| = 1, 000, 000.
20. Absolute deviation
The absolute deviation measures the average distance
from each value to the mean. It is another measure of
dispersion.
As a formula:
AD =
|x − ¯x|
n
21. Example
What is the absolute deviation of the data
0 2 0 1 0 8 2 1 0 0
The mean is ¯x = 1.4. We first work out |x − ¯x|:
1.4 0.6 1.4 0.4 1.4 6.6 0.6 0.4 1.4 1.4
The absolute deviation is
AD =
|x − ¯x|
n
=
15.6
10
= 1.56
22. Hypothesis testing
We use hypothesis testing to compare the mean of a very large
data set, a population mean, with the mean of a sample data
set, a sample mean.
Example: A lightbulb company says their lightbulbs last a mean
time of 1000 hours with a standard deviation of 50. We think
their lightbulbs last longer than this and propose a test at a 5%
level of significance. We buy 75 lightbulbs and they last a mean
time of 1022 hours.
The population mean is 1000 hours.
The sample is the 75 light bulbs that we test.
The sample mean is 1022 hours.
23. Hypothesis testing
The null hypothesis, H0 is a statement which is assumed to
be true.
Sample data is collected and tested to see if it is consistent
with the null hypothesis.
If the sample mean is significantly different from the
population mean, then we say that we have sufficient
evidence to reject the null hypothesis, H0, in favour of the
alternative hypothesis, H1.
24. The null hypothesis and the alternative hypothesis
The null hypothesis concerns the population mean.
It is of the form
H0 : µ = A
where µ is ’population mean’ and A is the hypothetical
value
The alternative hypothesis is that the null hypothesis is
incorrect and will be one of
H1 : µ = A
H1 : µ < A
H1 : µ > A
The question will direct you which of the above to use.
25. Significance level
The null hypothesis will always be tested to a given level of
significance.
A 5% level of significance means we are testing to see if
the probability of getting the sample data is less than 0.05.
If the probability is less we reject the null hypothesis in
favour of the alternative hypothesis.
A 1% level of significance translates to a probability of 0.01.
26. Critical value
A critical value is the value beyond which we reject the null
hypothesis. It tells us the boundary of the critical region(s)
In a Z-test this depends on the alternative hypothesis and
the significance level.
We look up the critical value(s) in tables.
Sig. Lev. 5% Sig. Lev. 1%
One-tail Two-tail One-tail Two-tail
Critical value 1.65 1.96 2.33 2.58
27. H1 : µ = A
If our alternative hypothesis is H1 : µ = A we are doing a
two-tailed test and we have 2 critical values, one negative and
one positive.
The critical value is the boundary of the rejection region.
For a 5% level of significance we have the following picture:
−1.96 1.96
x
y
The rejection (shaded) regions have a combined area of 0.05.
28. H1 : µ > A
If our alternative hypothesis is H1 : µ > A we are doing a
one-tailed test and we have 1 critical value which is positive.
The critical value is the boundary of the rejection region.
For a 5% level of significance we have the following picture:
1.65
x
y
The rejection region has an area of 0.05.
29. H1 : µ < A
If our alternative hypothesis is H1 : µ < A we are doing a
one-tailed test and we have 1 critical value which is negative.
The critical value is the boundary of the rejection region.
For a 5% level of significance we have the following picture:
1.65
x
y
The rejection region has an area of 0.05.
30. Test statistic
The test statistic is difference between the sample mean, ¯x
and the (hypothetical) population mean A, divided by the
standard error.
The standard error is σ/
√
n for the Z-test and s/
√
n for the
T-test, where n is the sample size, σ is the population
standard deviation and s is the sample standard deviation.
The Z-test statistic is
Z =
¯x − A
σ/
√
n
If the test statistic lies beyond the critical value(s) (in the
rejection region) we reject H0. If it does not, we accept H0.
31. Z-test - Example 1
Research says that the mean height for a man is 182cm with a
standard deviation of 9. We suspect men might be shorter than
this. We get the heights of 100 men and their mean height is
176. We test at a 1% level of significance.
32. Z-test - Example 1
The null hypothesis and alternative hypothesis are:
H0 : µ = 182
H1 : µ < 182
We are doing a 1-tailed test at a 1% level of significance so
the critical value is: C = −2.33.
The test statistic is Z = 176−182
9/
√
100
= −6.67.
−6.67 < −2.33 so we reject the null hypothesis.
33. Z-test - Example 2
A company says employees are supposed to work an average
of 40 hours a week with a standard deviation of 5 hours. Alfred
wants to know if he fits this to a 5% level of significance. He
notes down how many hours he works over 48 weeks and has
a mean of 39 hours.
34. Z-test - Example 2
The null hypothesis and alternative hypothesis are:
H0 : µ = 40
H1 : µ = 40
We are doing a 2-tailed test at a 5% level of significance so
the critical values are: C = −1.96, 1.96.
The test statistic is Z = 39−40
5/
√
48
= −1.39.
−1.96 < −1.39 < 1.96 so we accept the null hypothesis.
35. Z-test - Example 3
A lightbulb company says their lightbulbs last a mean time of
1000 hours with a standard deviation of 50. We think their
lightbulbs last longer than this and propose a test at a 5% level
of significance. We buy 75 lightbulbs and they last a mean time
of 1022 hours.
36. Z-test - Example 3
The null hypothesis and alternative hypothesis are:
H0 : µ = 1000
H1 : µ > 1000
We are doing a 1-tailed test at a 5% level of significance so
the critical value is: C = 1.65.
The test statistic is Z = 1022−1000
50/
√
75
= 3.81.
1.65 < 3.81 so we reject the null hypothesis.
37. Z-test summary
You will be given
1. Population mean, A
2. Population standard deviation, σ
3. Significance level
4. Sample mean, ¯x
5. Sample size, n
6. Quantifying word.
You have to work out
1. Null hypothesis, alternative hypotheis
2. Critical value(s)
3. Test statistic
4. Decision - accept/reject H0 (sketch a picture if possible)
5. Conclusion
38. The theory behind the Z-test and the T-test
If samples of size n are taken from a population with mean A
and standard deviation σ, then the sample means are
distributed normally, with mean A and standard deviation σ/
√
n
When we calculate the test statistic, we are calculating the
Z-score of the sample mean
The critical value is the Z-score of a sample mean which we
have a 5% (or 1%) probability of obtaining
For further information, try a statistics book from the library, or
the khanacademy videos on youtube
39. Normal distribution X ∼ N(µ, σ2
)
The normal distribution is defined as
f(x) =
1
σ
√
2π
e
−
(x−µ)2
2σ2
where σ is the population standard deviation and µ is the
population mean.
The graph below is when µ = 0 and σ = 1.
−4 −2 2 4
0.1
0.2
0.3
0.4
0.5
x
y
Probabilities correspond to areas under this curve