The document discusses testing hypotheses about the beta coefficients of technology shares. It provides:
1) Hypotheses to test if the average beta is greater than the market average of 1, using a one-tailed test at α=0.10.
2) Calculations to find the test statistic and critical value, showing the test statistic exceeds the critical value, so the null hypothesis is rejected.
3) A 95% confidence interval calculated for the population mean beta coefficient, ranging from 1.0251 to 1.4349.
4) A chi-square test showing the variance of the shares does not differ significantly from 0.15.
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Worked examples of sampling uncertainty evaluationGH Yeoh
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Common mistakes in measurement uncertainty calculationsGH Yeoh
The basic calculation for measurement uncertainty (MU) is through the law of propagation of uncertainty. Some find it difficult to apply and make some mistakes in the MU evaluation.
Hypothesis Testing: Central Tendency – Normal (Compare 1:1)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the 2 Sample T & Paired T tests as central tendency measurements for normal distributions.
Worked examples of sampling uncertainty evaluationGH Yeoh
ISO/IEC 17025:2017 laboratory accreditation standard has expanded its requirement for measurement uncertainty to include both sampling and analytical uncertainties.
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Chapter 8 Confidence Interval Estimation
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Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Traditional randomized experiments allow us to determine the overall causal impact of a treatment program (e.g. marketing, medical, social, education, political). Uplift modeling (also known as true lift, net lift, incremental lift) takes a further step to identify individuals who are truly positively influenced by a treatment through data mining / machine learning. This technique allows us to identify the “persuadables” and thus optimize target selection in order to maximize treatment benefits. This important subfield of data mining/data science/business analytics has gained significant attention in areas such as personalized marketing, personalized medicine, and political election with plenty of publications and presentations appeared in recent years from both industry practitioners and academics.
In this workshop, I will introduce the concept of Uplift, review existing methods, contrast with the traditional approach, and introduce a new method that can be implemented with standard software. A method and metrics for model assessment will be recommended. Our discussion will include new approaches to handling a general situation where only observational data are available, i.e. without randomized experiments, using techniques from causal inference. Additionally, an integrated modeling approach for uplift and direct response (where it can be identified who actually responded, e.g., click-through or coupon scanning) will be discussed. Last but not least, extension to the multiple treatment situation with solutions to optimizing treatments at the individual level will also be discussed. While the talk is geared towards marketing applications (“personalized marketing”), the same methodologies can be readily applied in other fields such as insurance, medicine, education, political, and social programs. Examples from the retail and non-profit industries will be used to illustrate the methodologies.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
Hypothesis Testing: Central Tendency – Normal (Compare 2+ Factors)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the ANOVA test as a central tendency measurement for normal distributions. It also explains what residuals and boxplots are and how to use them with the ANOVA test.
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Traditional randomized experiments allow us to determine the overall causal impact of a treatment program (e.g. marketing, medical, social, education, political). Uplift modeling (also known as true lift, net lift, incremental lift) takes a further step to identify individuals who are truly positively influenced by a treatment through data mining / machine learning. This technique allows us to identify the “persuadables” and thus optimize target selection in order to maximize treatment benefits. This important subfield of data mining/data science/business analytics has gained significant attention in areas such as personalized marketing, personalized medicine, and political election with plenty of publications and presentations appeared in recent years from both industry practitioners and academics.
In this workshop, I will introduce the concept of Uplift, review existing methods, contrast with the traditional approach, and introduce a new method that can be implemented with standard software. A method and metrics for model assessment will be recommended. Our discussion will include new approaches to handling a general situation where only observational data are available, i.e. without randomized experiments, using techniques from causal inference. Additionally, an integrated modeling approach for uplift and direct response (where it can be identified who actually responded, e.g., click-through or coupon scanning) will be discussed. Last but not least, extension to the multiple treatment situation with solutions to optimizing treatments at the individual level will also be discussed. While the talk is geared towards marketing applications (“personalized marketing”), the same methodologies can be readily applied in other fields such as insurance, medicine, education, political, and social programs. Examples from the retail and non-profit industries will be used to illustrate the methodologies.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
Solution to the practice test ch 8 hypothesis testing ch 9 two populationsLong Beach City College
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Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
1. ECF 6102 - Quantitative
Skills for Business
Tutorial / Research Paper C
Wednesday 13 May
15:30
ML14:114
Andrew Dash
2. What is Beta Coefficient?
• In finance evaluations the “beta coefficient” of a share is often considered a
measure of the stocks volatility (or in financial terms “risk”).
• Shares with beta coefficients greater than 1 generally bear greater risk, and
hence more volatility, than the overall market whereas
• Shares with beta coefficients less than 1 generally are considered less risky
(or less volatile) than the overall market.
• From previous studies it has been established that the “beta coefficient”
has been normally distributed.
3. Research Situation
Imagine we take a random sample of 15 technology shares at the end of 2014.
The mean and standard deviation of the beta coefficients for these data are
calculated as:
𝑿 = 𝟏. 𝟐𝟑 & 𝒔 = 𝟎. 𝟑𝟕
Other Solution Essentials
• 𝑛 = 15 • 𝛼 = 0.10 • 𝜇 = 1
Previous studies have shown that the “beta
coefficient” has been normally distributed.
4. Part A
Set up the appropriate null and alternate
hypotheses to test whether the average
technology shares are riskier than the
market as a whole.
5. One orTwo tailed?
• Keywords in the question are‘…riskier than the market as a whole’.
• In other words, is the average beta coefficient of our sample greater than
the beta coefficient of the market as a whole?
• Therefore, it is a one tailed test.
𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1
The Golden Rule
𝑋 > 𝜇 = 𝑈𝑝𝑝𝑒𝑟 𝑇𝑎𝑖𝑙 𝑇𝑒𝑠𝑡
𝑋 < 𝜇 = 𝐿𝑜𝑤𝑒𝑟 𝑇𝑎𝑖𝑙 𝑇𝑒𝑠𝑡
• As 𝑋 (1.23) is greater than 𝜇 (1), it is an upper tailed test.
6. The null & alternate hypotheses
• Setup the alternate hypothesis (𝐻1) first.
• The alternate hypothesis is setup to answer the question posed.
• Therefore:
𝐻0: 𝜇 ≤ 1
𝐻1: 𝜇 > 1
𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1
7. Part B
Using your sampling decision tree,
establish the appropriate test statistic
and rejection region for the test using
alpha = 0.10.
11. The Decision Rule
• Our hypothesis: 𝐻0: 𝜇 ≤ 1
𝐻1: 𝜇 > 1
• Decision Rule: Reject 𝐻0 if 𝑡test > +1.3450, otherwise do not reject 𝐻0.
Do not reject H0
0
Reject H0
t
α=0.10
𝑡 𝑐𝑣 = +1.3450
𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
Image adapted from example provided by M Waring (personal communication, 30 March 2015)
12. Testing our hypothesis
• Our test statistic: 𝑡𝑡𝑒𝑠𝑡 =
𝑋−𝜇
𝑠 𝑋
=
1.23−1
0.09553
= +2.4075
Do not reject H0
0
Reject H0
t
𝑡 𝑐𝑣 = +1.3450
𝑡𝑡𝑒𝑠𝑡 = +2.4075
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
Image adapted from example provided by M Waring (personal communication, 30 March 2015)
13. Our Conclusion
• Since 𝑡𝑡𝑒𝑠𝑡 2.4075 > 𝑡 𝑐𝑣(1.3450), we reject 𝐻0 as there is enough
evidence to conclude that the average technology shares are riskier than the
market as a whole.
Null Hypothesis m= 1
Level of Significance 0.1
Sample Size 15
Sample Mean 1.23
Sample Standard Deviation 0.37
Standard Error of the Mean 0.0955
Degrees of Freedom 14
t Test Statistic 2.4075
Upper-Tail Test Calculations Area
Upper Critical Value 1.3450
p-Value 0.0152
Reject the null hypothesis
Data
Intermediate Calculations
t Test for Hypothesis of the Mean
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
14. Part D
Use PhStat to determine the approximate p-
value associated with this test.
What does the p-value theoretically mean?
15. PhStat and the p-Value
• “The p-value is the probability of getting a test statistic equal to or more
extreme than the sample result, given that the null hypothesis, 𝐻0 is
true.The p-value is also known as the observed level of significance”
(Berenson, Levine, & Krehbiel, 2012).
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
16. PhStat and the p-Value
• The strength of the decision concerning H0 is found by
comparing the p-value to the alpha () level.
• If p-Value is low, 𝐻0 must go…
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
Null Hypothesis m= 1
Level of Significance 0.1
Sample Size 15
Sample Mean 1.23
Sample Standard Deviation 0.37
Standard Error of the Mean 0.0955
Degrees of Freedom 14
t Test Statistic 2.4075
Upper-Tail Test
Upper Critical Value 1.345030374
p-Value 0.015213385
Reject the null hypothesis
Data
Intermediate Calculations
t Test for Hypothesis of the Mean
Our Data
• p-Value (0.0152) < 𝛼 (0.1).
• Therefore we reject 𝐻0.
17. PhStat and the p-Value
• The p-Value is an important result because it measures the amount of
statistical evidence that supports the alternative hypothesis.
• A small p-Value indicates that there is ample evidence to support the
alternative hypothesis.
• A large p-Value indicates that there is little evidence to support the
alternative hypothesis.
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
18. Part E
If we had tested this same hypothesis
using the Z distribution at alpha = 0.10
would you have drawn the same
conclusion?
20. Testing our hypothesis
• Our decision rule: Reject 𝐻0 if 𝑍𝑡𝑒𝑠𝑡 +2.4075 > 𝑍 𝑐𝑣(+1.28), otherwise
do not reject 𝐻0
Do not reject H0
0
Reject H0
t
𝑍 𝑐𝑣 = +1.28
𝑍𝑡𝑒𝑠𝑡 = +2.4075
𝑍𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑍 𝑐𝑣 = +1.28
Image adapted from example provided by M Waring (personal communication, 30 March 2015)
21. Our Conclusion
• Since 𝑍𝑡𝑒𝑠𝑡 2.4075 > 𝑍 𝑐𝑣(1.28), therefore we reject 𝐻0 as there is enough
evidence to conclude that the average technology shares are riskier than the
market as a whole.
𝑍𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑍 𝑐𝑣 = +1.28
Null Hypothesis m= 1
Level of Significance 0.1
Population Standard Deviation 0.37
Sample Size 15
Sample Mean 1.23
Standard Error of the Mean 0.0955
Z Test Statistic 2.4075
Upper-Tail Test
Upper Critical Value 1.2816
p-Value 0.0080
Reject the null hypothesis
Data
Intermediate Calculations
Z Test of Hypothesis for the Mean
22. Part F
Develop a 95% confidence interval for the
“beta coefficient”.
24. CriticalValues of t
Therefore, 𝑡 𝛼/2, 𝑑. 𝑓. = 𝑡0.025; 14 = 2.1448
𝑛 = 15 𝑥 = 1.23 𝑠 = 0.37 1 − 𝛼 = 95% 𝑡0.025; 14 = ±2.1448
(M Waring , personal communication, March 2015)
25. • The formula for the confidence interval estimation of the mean is:
𝑋 − 𝑡 𝛼 2
𝑠
𝑛
≤ 𝜇 ≤ 𝑋 + 𝑡 𝛼 2
𝑠
𝑛
The Calculations
.5000 .5000
0.95
(1-)
.4750 .4750
/2= 0.025 /2=0.025
s x
mm -2.1448 m +2.1448 s x
t-2.1448 0 2.1448
𝑛 = 15 𝑥 = 1.23 𝑠 = 0.37 1 − 𝛼 = 95% 𝑡0.025; 14 = ±2.1448
Image adapted from example provided by M Waring (personal communication, 30 March 2015)
26. Therefore, we can be 95% confident that the mean beta coefficient lies between 1.0251
and 1.4349
The Calculations
1.23 − 2.1448
0.37
15
≤ 𝜇 ≤ 1.23 + 2.1448
0.37
15
𝑋 − 𝑡 𝛼 2
𝑠
𝑛
≤ 𝜇 ≤ 𝑋 + 𝑡 𝛼 2
𝑠
𝑛
1.0251 ≤ 𝜇 ≤ 1.4349
1.23 − 0.2049 ≤ 𝜇 ≤ 1.23 + 0.2049
𝑛 = 15 𝑥 = 1.23 𝑠 = 0.37 1 − 𝛼 = 95% 𝑡0.025; 14 = ±2.1448
27. Confidence Interval Estimate for the Mean
Data
Sample Standard Deviation 0.37
Sample Mean 1.23
Sample Size 15
Confidence Level 95%
Standard Error of the Mean 0.095533589
Degrees of Freedom 14
t Value 2.1448
Interval Half Width 0.2049
Interval Lower Limit 1.0251
Interval Upper Limit 1.4349
Intermediate Calculations
Confidence Interval
Just in case…
1.0251 ≤ 𝜇 ≤ 1.4349
𝑛 = 15 𝑥 = 1.23 𝑠 = 0.37 1 − 𝛼 = 95% 𝑡0.025; 14 = ±2.1448
28. Part G
Use PhStat software to conduct a test to
determine if the variance of the shares beta
value differs from 0.15 at alpha = 0.05.
Show this printout
29. Chi-SquareTest ofVariance
Data
Null Hypothesis s^2= 0.15
Level of Significance 0.05
Sample Size 15
Sample Standard Deviation 0.37
Degrees of Freedom 14
Half Area 0.025
Chi-Square Statistic 12.7773
Two-Tail Test
Lower Critical Value 5.6287
Upper Critical Value 26.1189
p-Value 0.4559
Do not reject the null hypothesis
Intermediate Calculations
Chi-Square Test of Variance
31. The Confirmation
Since 𝑡𝑡𝑒𝑠𝑡 2.4075 > 𝑡 𝑐𝑣(1.3450),
we reject 𝐻0 as there is enough
evidence to conclude that the
average technology shares are
riskier than the market as a whole.
Null Hypothesis m= 1
Level of Significance 0.1
Sample Size 15
Sample Mean 1.23
Sample Standard Deviation 0.37
Standard Error of the Mean 0.0955
Degrees of Freedom 14
t Test Statistic 2.4075
Upper-Tail Test Calculations Area
Upper Critical Value 1.3450
p-Value 0.0152
Reject the null hypothesis
Data
Intermediate Calculations
t Test for Hypothesis of the Mean
𝑡𝑡𝑒𝑠𝑡 = +2.4075𝑛 = 15 𝑋 = 1.23 𝑠 = 0.37 𝛼 = 0.10 𝜇 = 1 𝑡 𝑐𝑣 = +1.3450
32. References
• Mark L. Berenson,T. C. K. D. M. L. (2012). Basic Business Statistics:
Concepts and Applications (E. Svendsen Ed. Vol. 12): Pearson
Education, Inc.