This document provides an overview of hypothesis testing. It defines key terms like the null hypothesis (H0), alternative hypothesis (H1), type I and type II errors, significance level, p-values, and test statistics. It explains the basic steps in hypothesis testing as testing a claim about a population parameter by collecting a sample, determining the appropriate test statistic based on the sampling distribution, and comparing it to critical values to reject or fail to reject the null hypothesis. Examples are provided to demonstrate hypothesis tests for a mean when the population standard deviation is known or unknown.
Hypothesis Testing is important part of research, based on hypothesis testing we can check the truth of presumes hypothesis (Research Statement or Research Methodology )
Hypothesis Testing is important part of research, based on hypothesis testing we can check the truth of presumes hypothesis (Research Statement or Research Methodology )
Hypothesis testing and estimation are used to reach conclusions about a population by examining a sample of that population.
Hypothesis testing is widely used in medicine, dentistry, health care, biology and other fields as a means to draw conclusions about the nature of populations
Chi-Square test for independence of attributes / Chi-Square test for checking association between two categorical variables, Chi-Square test for goodness of fit
Test statistic
General Formula for Test Statistic
Statistical Decision
Significance Level
p-values
Purpose of Hypothesis Testing
General Procedure of Testing a Hypothesis
Hypothesis testing and estimation are used to reach conclusions about a population by examining a sample of that population.
Hypothesis testing is widely used in medicine, dentistry, health care, biology and other fields as a means to draw conclusions about the nature of populations
Chi-Square test for independence of attributes / Chi-Square test for checking association between two categorical variables, Chi-Square test for goodness of fit
Test statistic
General Formula for Test Statistic
Statistical Decision
Significance Level
p-values
Purpose of Hypothesis Testing
General Procedure of Testing a Hypothesis
Hypothesis Testing
(Statistical Significance)
1
Hypothesis Testing
Goal: Make statement(s) regarding unknown population parameter values based on sample data
Elements of a hypothesis test:
Null hypothesis - Statement regarding the value(s) of unknown parameter(s). Typically will imply no association between explanatory and response variables in our applications (will always contain an equality)
Alternative hypothesis - Statement contradictory to the null hypothesis (will always contain an inequality)
The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha
Definitions
Rejection (alpha, α) Region:
Represents area under the curve that is used to reject the null hypothesis
Level of Confidence, 1 - alpha (a):
Also known as fail to reject (FTR) region
Represents area under the curve that is used to fail to reject the null hypothesis
FTR
H0
α/2
α/2
3
1 vs. 2 Sided Tests
Two-sided test
No a priori reason 1 group should have stronger effect
Used for most tests
Example
H0: μ1 = μ2
HA: μ1 ≠ μ2
One-sided test
Specific interest in only one direction
Not scientifically relevant/interesting if reverse situation true
Example
H0: μ1 ≤ μ2
HA: μ1 > μ2
4
Example: It is believed that the mean age of smokers in San Bernardino is 47. Researchers from LLU believe that the average age is different than 47.
Hypothesis
H0:μ = 47
HA: μ ≠ 47
μ = 47
α /2 = 0.025
Fail to Reject (FTR)
α /2 = 0.025
5
Three Approaches to Reject or Fail to Reject A Null Hypothesis:
1a. Confidence interval
Calculate the confidence interval
Decision Rule:
a. If the confidence interval (CI) includes the null, then the decision must be to fail to reject the H0.
b. If the confidence interval (CI) does not include the null, then the decision must be to reject the H0.
6
1b. Confidence interval to compare groups
Calculate the confidence interval for each group
Decision Rule:
a. If the confidence interval (CI) overlap, then the decision must be to fail to reject the H0.
b. If the confidence interval (CI) do not include the null, then the decision must be to reject the H0.
7
2.Test Statistic
Calculate the test statistic (TS)
Obtain the critical value (CV) from the reference table
Decision Rule:
a. If the test statistic is in the FTR region, then the decision must be to fail to reject the H0.
b. If the test statistic is in the rejection region, then the decision must be to reject the H0.
FTR
CV
TS
Since the test statistic is in the rejection region, reject the H0
FTR
CV
Since the test statistic is in the fail to reject region, fail to reject the H0
TS
CV
CV
8
3. P-Value
Choose α
Calculate value of test statistic from your data
Calculate P- value from test statistic
Decision Rule:
a. If the p-value is less than the level of significance, α, then the decision m.
Int 150 The Moral Instinct”1. Most cultures agree that abus.docxmariuse18nolet
Int 150
“The Moral Instinct”
1. Most cultures agree that abusing innocent people is wrong. True or false
2. Young children have a sense of morality. True or false (example)?
3. Emotional reasoning trumps rationalizing. True or false (explain)
4. According to the article, psychopathy or moral misbehavior (like rape) is more environmental than genetic. True or false (example)
5. Explain the point about the British schoolteacher in Sudan.
6. Name three things anthropologists believe all people share, in addition to thinking it’s bad to harm others and good to help them.
a.
b.
c.
7. What is reciprocal altruism?
8. How does the psychologist Tetlock explain the outrage of American college students at the thought that adoption agencies should place children with couples willing to pay the most?
9. Discuss: A love for children and sense of justice is just an expression of our innate sense of preserving our genes for future generations (Darwin)
10. What does the author warn about the arguments regarding climate change?
Hypothesis Testing
(Statistical Significance)
1
Hypothesis Testing
Goal: Make statement(s) regarding unknown population parameter values based on sample data
Elements of a hypothesis test:
Null hypothesis - Statement regarding the value(s) of unknown parameter(s). Typically will imply no association between explanatory and response variables in our applications (will always contain an equality)
Alternative hypothesis - Statement contradictory to the null hypothesis (will always contain an inequality)
The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha
Definitions
Rejection (alpha, α) Region:
Represents area under the curve that is used to reject the null hypothesis
Level of Confidence, 1 - alpha (a):
Also known as fail to reject (FTR) region
Represents area under the curve that is used to fail to reject the null hypothesis
FTR
H0
α/2
α/2
3
1 vs. 2 Sided Tests
Two-sided test
No a priori reason 1 group should have stronger effect
Used for most tests
Example
H0: μ1 = μ2
HA: μ1 ≠ μ2
One-sided test
Specific interest in only one direction
Not scientifically relevant/interesting if reverse situation true
Example
H0: μ1 ≤ μ2
HA: μ1 > μ2
4
Example: It is believed that the mean age of smokers in San Bernardino is 47. Researchers from LLU believe that the average age is different than 47.
Hypothesis
H0:μ = 47
HA: μ ≠ 47
μ = 47
α /2 = 0.025
Fail to Reject (FTR)
α /2 = 0.025
5
Three Approaches to Reject or Fail to Reject A Null Hypothesis:
1a. Confidence interval
Calculate the confidence interval
Decision Rule:
a. If the confidence interval (CI) includes the null, then the decision must be to fail to reject the H0.
b. If the confidence interval (CI) does not include the null, then the decision must be to reject the H0.
6
1b. Confidence interval to compare groups
Calculate the confidence interval for each gro.
[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.
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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.
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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.
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
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Data Centers - Striving Within A Narrow Range - Research Report - MCG - May 2...pchutichetpong
M Capital Group (“MCG”) expects to see demand and the changing evolution of supply, facilitated through institutional investment rotation out of offices and into work from home (“WFH”), while the ever-expanding need for data storage as global internet usage expands, with experts predicting 5.3 billion users by 2023. These market factors will be underpinned by technological changes, such as progressing cloud services and edge sites, allowing the industry to see strong expected annual growth of 13% over the next 4 years.
Whilst competitive headwinds remain, represented through the recent second bankruptcy filing of Sungard, which blames “COVID-19 and other macroeconomic trends including delayed customer spending decisions, insourcing and reductions in IT spending, energy inflation and reduction in demand for certain services”, the industry has seen key adjustments, where MCG believes that engineering cost management and technological innovation will be paramount to success.
MCG reports that the more favorable market conditions expected over the next few years, helped by the winding down of pandemic restrictions and a hybrid working environment will be driving market momentum forward. The continuous injection of capital by alternative investment firms, as well as the growing infrastructural investment from cloud service providers and social media companies, whose revenues are expected to grow over 3.6x larger by value in 2026, will likely help propel center provision and innovation. These factors paint a promising picture for the industry players that offset rising input costs and adapt to new technologies.
According to M Capital Group: “Specifically, the long-term cost-saving opportunities available from the rise of remote managing will likely aid value growth for the industry. Through margin optimization and further availability of capital for reinvestment, strong players will maintain their competitive foothold, while weaker players exit the market to balance supply and demand.”
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
2. What is a Hypothesis?
A hypothesis is a claim
(assumption) about a
population parameter:
population mean
population proportion
Example: The mean monthly cell phone bill
in this city is μ = $42
Example: The proportion of adults in this
city with cell phones is π = 0.68
3. The Null Hypothesis, H0
States the claim or assertion to be tested
Example: The average number of TV sets in
U.S. Homes is equal to three ( )
Is always about a population parameter,
not about a sample statistic
3μ:H0 =
3μ:H0 = 3X:H0 =
4. The Null Hypothesis, H0
Begin with the assumption that the null
hypothesis is true
Similar to the notion of innocent until
proven guilty
Refers to the status quo or historical value
Always contains “=” , “≤” or “≥” sign
May or may not be rejected
(continued)
5. The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g., The average number of TV sets in U.S.
homes is not equal to 3 ( H1: μ ≠ 3 )
Challenges the status quo
May or may not be proven
Is generally the hypothesis that the
researcher is trying to prove
6. The Hypothesis Testing
Process
Claim: The population mean age is 50.
H0: μ = 50, H1: μ ≠ 50
Sample the population and find sample mean.
Population
Sample
7. The Hypothesis Testing
Process
Sampling
Distribution of X
μ = 50
If H0 is true
If it is unlikely that you
would get a sample
mean of this value ...
... then you reject
the null hypothesis
that μ = 50.
20
... When in fact this were
the population mean…
X
(continued)
11. The Test Statistic and
Critical Values
Critical Values
“Too Far Away” From Mean of Sampling Distribution
(Significant)
Sampling Distribution of the test statistic
Region of
Rejection
Region of
Rejection
Region of
Non-Rejection
12. Possible ErrorsErrors in Hypothesis Test
Type I Error
Reject a true null hypothesis
Considered a serious type of error
The probability of a Type I Error is α
Called level of significance of the test
Set by researcher in advance
Type II Error
Failure to reject false null hypothesis
The probability of a Type II Error is β
13. Possible Errors in Hypothesis Test
Decision Making
Possible Hypothesis Test Outcomes
Actual Situation
Decision H0 True H0 False
Do Not
Reject H0
No Error
Probability 1 - α
Type II Error
Probability β
Reject H0 Type I Error
Probability α
No Error
Probability 1 - β
(continued)
14. Possible Errors in Hypothesis Test
Decision Making
The confidence coefficient (1-α) is the
probability of not rejecting H0 when it is true.
The confidence level of a hypothesis test is
(1-α)*100%.
The power of a statistical test (1-β) is the
probability of rejecting H0 when it is false.
(continued)
15. Type I & II Error Relationship
Type I and Type II errors cannot happen at
the same time
A Type I error can only occur if H0 is true
A Type II error can only occur if H0 is false
If Type I error probability ( α ) , then
Type II error probability ( β )
16. Level of Significance
and the Rejection Region
Level of significance = α
This is a two-tail test because there is a rejection region in both tails
H0: μ = 3
H1: μ ≠
3
Critical values
Rejection Region
/2
0
α/2α
17. Hypothesis Tests for the Meanfor the Mean
σσ KnownKnown σσ UnknownUnknown
Hypothesis
Tests for µ
(Z test) (t test)
18. Z Test of Hypothesis for the
Mean (σ Known)
Convert sample statistic ( ) to a ZSTAT test statisticX
The test statistic is:
n
σ
μX
STATZ
−
=
σ Known σ Unknown
Hypothesis
Tests for µ
σ Known σ Unknown
(Z test) (t test)
19. Critical Value
Approach to Testing
For a two-tail test for the mean, σ known:
Convert sample statistic ( ) to test statistic
(ZSTAT)
Determine the critical Z values for a specified
level of significance α from a table
Decision Rule: If the test statistic falls in the
rejection region, reject H0 ; otherwise do not
reject H0
X
20. Do not reject H0 Reject H0Reject H0
There are two
cutoff values
(critical values),
defining the
regions of
rejection
Two-Tail Tests
α/2
-Zα/2 0
H0: μ = 3
H1: μ ≠
3
+Zα/2
α/2
Lower
critical
value
Upper
critical
value
3
Z
X
21. 6 Steps in
Hypothesis Testing
1. State the null hypothesis, H0 and the
alternative hypothesis, H1
2. Choose the level of significance, α, and the
sample size, n
3. Determine the appropriate test statistic and
sampling distribution
4. Determine the critical values that divide the
rejection and non-rejection regions
22. 6 Steps in
Hypothesis Testing
5. Collect data and compute the value of the test
statistic
6. Make the statistical decision and state the
managerial conclusion. If the test statistic falls
into the non-rejection region, do not reject the
null hypothesis H0. If the test statistic falls into
the rejection region, reject the null hypothesis.
Express the managerial conclusion in the
context of the problem
(continued)
23. Hypothesis Testing Example
Test the claim that the true mean # of
TV sets in US homes is equal to 3.
(Assume σ = 0.8)
1. State the appropriate null and alternative
hypotheses
H0: μ = 3 H1: μ ≠ 3 (This is a two-tail test)
2. Specify the desired level of significance and the
sample size
Suppose that α = 0.05 and n = 100 are chosen
for this test
24. 2.0
.08
.16
100
0.8
32.84
n
σ
μX
STATZ −=
−
=
−
=
−
=
Hypothesis Testing Example
3. Determine the appropriate technique
σ is known so this is a Z test.
4. Determine the critical values
For α = 0.05 the critical Z values are ±1.96
5. Collect the data and compute the test statistic
Suppose the sample results are
n = 100, X = 2.84 (σ = 0.8 is assumed known)
So the test statistic is:
(continued)
25. Reject H0 Do not reject H0
6. Is the test statistic in the rejection region?
α/2 = 0.025
-Zα/2 = -1.96 0
Reject H0 if
ZSTAT < -1.96 or
ZSTAT > 1.96;
otherwise do
not reject H0
Hypothesis Testing Example
(continued)
α/2 = 0.025
Reject H0
+Zα/2 = +1.96
Here, ZSTAT = -2.0 < -1.96, so the
test statistic is in the rejection
region
26. 6 (continued). Reach a decision and interpret the result
-2.0
Since ZSTAT = -2.0 < -1.96, reject the null hypothesis
and conclude there is sufficient evidence that the mean
number of TVs in US homes is not equal to 3
Hypothesis Testing Example
(continued)
Reject H0 Do not reject H0
α = 0.05/2
-Zα/2 = -1.96 0
α = 0.05/2
Reject H0
+Zα/2= +1.96
29. p-Value Approach to Testing
p-value: Probability of obtaining a test
statistic equal to or more extreme than the
observed sample value given H0 is true
The p-value is also called the observed level of
significance
It is the smallest value of α for which H0 can be
rejected
30. p-Value Approach to Testing:
Interpreting the p-value
Compare the p-value with α
If p-value < α , reject H0
If p-value ≥ α , do not reject H0
Remember
If the p-value is low then H0 must go
31. p-value Hypothesis Testing
Example
Test the claim that the true mean # of
TV sets in US homes is equal to 3.
(Assume σ = 0.8)
1. State the appropriate null and alternative
hypotheses
H0: μ = 3 H1: μ ≠ 3 (This is a two-tail test)
2. Specify the desired level of significance and the
sample size
Suppose that α = 0.05 and n = 100 are chosen
for this test
32. 2.0
.08
.16
100
0.8
32.84
n
σ
μX
STATZ −=
−
=
−
=
−
=
p-value Hypothesis Testing
Example
3. Determine the appropriate technique
σ is assumed known so this is a Z test.
4. Collect the data, compute the test statistic and the
p-value
Suppose the sample results are
n = 100, X = 2.84 (σ = 0.8 is assumed known)
So the test statistic is:
(continued)
33. p-Value Hypothesis Testing Example:
Calculating the p-value
4. (continued) Calculate the p-value.
How likely is it to get a ZSTAT of -2 (or something further from the
mean (0), in either direction) if H0 is true?
p-value = 0.0228 + 0.0228 = 0.0456
P(Z < -2.0) = 0.0228
0
-2.0
Z
2.0
P(Z > 2.0) = 0.0228
34. 5. Is the p-value < α?
Since p-value = 0.0456 < α = 0.05 Reject H0
5. (continued) State the managerial conclusion
in the context of the situation.
There is sufficient evidence to conclude the average number of
TVs in US homes is not equal to 3.
p-value Hypothesis Testing
Example
(continued)
35. Connection Between Two Tail
Tests and Confidence Intervals
For X = 2.84, σ = 0.8 and n = 100, the 95%
confidence interval is:
2.6832 ≤ μ ≤ 2.9968
Since this interval does not contain the hypothesized
mean (3.0), we reject the null hypothesis at α = 0.05
100
0.8
(1.96)2.84to
100
0.8
(1.96)-2.84 +
36. Hypothesis Testing:
σ Unknown
If the population standard deviation is unknown, you
instead use the sample standard deviation S.
Because of this change, you use the t distribution instead
of the Z distribution to test the null hypothesis about the
mean.
When using the t distribution you must assume the
population you are sampling from follows a normal
distribution.
All other steps, concepts, and conclusions are the
same.
37. t Test of Hypothesis for the Mean
(σ Unknown)
The test statistic is:
n
S
μX
STATt
−
=
Hypothesis
Tests for µ
σ Known σ Unknownσ Known σ Unknown
(Z test) (t test)
t Test of Hypothesis for the Mean
(σ Unknown)
Convert sample statistic ( ) to a tSTAT test statistic
The test statistic is:
n
S
μX
STATt
−
=
Hypothesis
Tests for µ
σ Known σ Unknownσ Known σ Unknown
(Z test) (t test)
X
The test statistic is:
n
S
μX
STATt
−
=
Hypothesis
Tests for µ
σ Known σ Unknownσ Known σ Unknown
(Z test) (t test)
39. Example: Two-Tail Test
(σ Unknown)
The average cost of a hotel
room in New York is said to
be $168 per night. To
determine if this is true, a
random sample of 25 hotels
is taken and resulted in an X
of $172.50 and an S of
$15.40. Test the appropriate
hypotheses at α = 0.05.
(Assume the population distribution is normal)
H0: μ = 168
H1: μ ≠
168
40. α = 0.05
n = 25, df = 25-1=24
σ is unknown, so
use a t statistic
Critical Value:
±t24,0.025 = ± 2.0639
Example Solution:
Two-Tail t TestTwo-Tail t Test
Do not reject H0: insufficient evidence that true
mean cost is different than $168
Reject H0Reject H0
α/2=.025
-t 24,0.025
Do not reject H0
0
α/2=.025
-2.0639 2.0639
1.46
25
15.40
168172.50
n
S
μX
STATt =
−
=
−
=
1.46
H0: μ = 168
H1: μ ≠
168
t 24,0.025
41. Connection of Two Tail Tests to
Confidence Intervals
For X = 172.5, S = 15.40 and n = 25, the 95%
confidence interval for µ is:
172.5 - (2.0639) 15.4/ 25 to 172.5 + (2.0639) 15.4/ 25
166.14 ≤ μ ≤ 178.86
Since this interval contains the Hypothesized mean (168),
we do not reject the null hypothesis at α = 0.05
42. One-Tail Tests
In many cases, the alternative hypothesis
focuses on a particular direction
H0: μ ≥ 3
H1: μ < 3
H0: μ ≤ 3
H1: μ > 3
This is a lower-tail test since the
alternative hypothesis is focused on
the lower tail below the mean of 3
This is an upper-tail test since the
alternative hypothesis is focused on
the upper tail above the mean of 3
43. Reject H0 Do not reject H0
There is only one
critical value, since
the rejection area is
in only one tail
Lower-Tail Tests
α
-Zα or -tα
0
μ
H0: μ ≥ 3
H1: μ < 3
Z or t
X
Critical value
44. Reject H0Do not reject H0
Upper-Tail Tests
α
Zα or tα0
μ
H0: μ ≤ 3
H1: μ > 3
There is only one
critical value, since
the rejection area is
in only one tail
Critical value
Z or t
X
_
45. Example: Upper-Tail t Test
for Mean (σ unknown)
A telecom manager thinks that customer monthly
cell phone bills have increased, and now
average over $52 per month. The company
wishes to test this claim. (Assume a normal
population)
H0: μ ≤ 52 the average is not over $52 per month
H1: μ > 52 the average is greater than $52 per month
(i.e., sufficient evidence exists to support the
manager’s claim)
Form hypothesis test:
46. Reject H0Do not reject H0
Suppose that α = 0.10 is chosen for this test and
n = 25.
Find the rejection region:
α = 0.10
1.3180
Reject H0
Reject H0 if tSTAT > 1.318
Example: Find Rejection Region
(continued)
47. Obtain sample and compute the test statistic
Suppose a sample is taken with the following
results: n = 25, X = 53.1, and S = 10
Then the test statistic is:
0.55
25
10
5253.1
n
S
μX
tSTAT
=
−
=
−
=
Example: Test Statistic
(continued)
48. Reject H0Do not reject H0
Example: Decision
α = 0.10
1.318
0
Reject H0
Do not reject H0 since tSTAT = 0.55 ≤ 1.318
there is not sufficient evidence that the
mean bill is over $52
tSTAT = 0.55
Reach a decision and interpret the result:
(continued)
49. Hypothesis Tests for ProportionsProportions
Involves categorical variables
Two possible outcomes
Possesses characteristic of interest
Does not possess characteristic of interest
Fraction or proportion of the population in the
category of interest is denoted by π
50. ProportionsProportions
Sample proportion in the category of interest is
denoted by p
When both nπ and n(1-π) are at least 5, p can
be approximated by a normal distribution with
mean and standard deviation
sizesample
sampleininterestofcategoryinnumber
n
X
p ==
π=pμ
n
)(1
σ
ππ −
=p
(continued)
51. The sampling
distribution of p is
approximately
normal, so the test
statistic is a ZSTAT
value:
Hypothesis Tests for Proportions
n
)(1
p
STATZ
ππ
π
−
−
=
nπ ≥ 5
&
n(1-π) ≥ 5
52. Example: Z Test for Proportion
A marketing company
claims that it receives
8% responses from its
mailing. To test this
claim, a random sample
of 500 were surveyed
with 25 responses. Test
at the α = 0.05
significance level.
Check:
nπ = (500)(.08) = 40
n(1-π) = (500)(.92) = 460
53. Z Test for Proportion: Solution
α = 0.05
n = 500, p = 0.05
Reject H0 at α = 0.05
H0: π = 0.08
H1: π ≠
0.08
Critical Values: ± 1.96
Test Statistic:
Decision:
Conclusion:
z0
Reject Reject
.025.025
1.96
-2.47
There is sufficient
evidence to reject the
company’s claim of 8%
response rate.
2.47
500
.08).08(1
.08.05
n
)(1
p
STATZ −=
−
−
=
−
−
=
ππ
π
-1.96
62. Paired t TestPaired t Test (Dependent Samples)(Dependent Samples)
Paired t test is used when experiments are conducted on the same set
of inputs (in this case ‘locations’)
66. Chi-squaredChi-squared distributiondistribution
The chi-squared distribution (also chi-square or χ²-distribution) with k (or sometimes ‘df’)
degrees of freedom is the distribution of a sum of the squares of k independent standard
normal random variables.
If Z1, ..., Zk are independent, standard normal random variables, then the sum of their
squares,
is distributed according to the chi-squared distribution with k degrees of freedom. This is
usually denoted as
Use: It helps us understand the relationship between categorical variables like age, sex,
year etc.
68. SolutionSolution
Similarly, we can fill the ‘expected values’ of other cells. General formula is
EV = (Row Total*Column total)/Grand total; here it is 25*60/100 = 15
70. SolutionSolution
Now look up the chi squared table for 1 degree of freedom & 5% significance.
The value is 3.841; since 2 < 3.841, we can conclude that there is no
significant difference between the two genders.
74. The F TableThe F Table
There is one table for each level of significance!
DF in Numerator (n-1)
DF in Denominator
(n-1)
75. Calculating the F StatisticCalculating the F Statistic
Highest variance/Lowest variance
(across all the samples)
76. ConcludingConcluding
F value = ___________
Critical value (from the F-table for .05 significance) = _____
If F-value < Critical Value => DO NOT reject the null hypothesis,
that is, the variances across samples are the same.