Tutorial 5 1. Suppose we have n = 10 observations (Xi , Yi ) and t the data with model Yi = 0 + 1 Xi + i with i , i = 1, ., 10 are IID N (0, 2 ). We have the following calculations.
n
X = 0.5669,
n i=1
Y = 0.9624,
i=1 n
Yi2 = 10.2695,
Xi2 = 4.0169,
Xi Y
TUTORIAL 3 1. Refer to the Grade point average problem (see tutorial 1) (a) obtain a 95% percent interval estimate of the mean freshman GPA for students whose ACT test score is 28. Interpret your condence interval. (b) Mary Jones obtained a score of 28 on
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Chapter 10
Comparing Two Groups
Bivariate Analyses: A Response Variable
and a Binary Explanatory Variable
Methods for comparing two groups are special cases of
bivariate statistical methods.
The outcome variable on which comparisons are
made is the respo
Chapter 3
Association: Contingency,
Correlation, and Regression
Section 3.1
The Association Between
Two Categorical Variables
Response and Explanatory Variables
Response variable (Dependent Variable)
The outcome variable on which comparisons are made.
Exp
Chapter 5
Probability
5.1 How Probability Quantifies Randomness
5.2 Finding Probabilities
5.3 Conditional Probabilities
5.4 Applying the Probability Rules
Random Phenomena
For random phenomena, the outcome is uncertain.
In the short-run, the proportion o
Chapter 6
Probability Distributions
Section 6.1
Summarizing Possible Outcomes and Their
Probabilities
1
Randomness
A random variable is a numerical measurement of the
outcome of a random phenomenon.
Often, the randomness results from
selecting a random s
Recitation Week 12
Problems: 9-10; 9-19; 9-26; 9-28; 9-43; 9-58.
9-10. A wall clock on Planet X has two hands that are aligned at midnight and turn in the
same direction at uniform rates, one at 0.0435 rad/s and the other at 0.0163 rad/s. At how
many seco
NATIONAL UNIVERSITY OF SINGAPORE
Department of Statistics and Applied Probability
2016/17 Semester 2
1.
ST3131 Regression Analysis
Tutorial 8
The table below gives the systolic blood pressure (y), body size (x1), age (x2), and
smoking history (x3 = 0 for
ST3131 (2016/2017 Semester 2) Partial Solutions/Hints to Questions in Tutorial 3
Note: The solutions provided in this document are for reference only.
Question 1
(i) The model is given by
= 0 + 1 () + 2 () + 3 () +
The fitted model is given by
= 262.50
NATIONAL UNIVERSITY OF SINGAPORE
Department of Statistics and Applied Probability
2016/17 Semester 1
1.
ST3131 Regression Analysis
Tutorial 3
The table below provides data on infant mortality (infant deaths per 1000 live births),
literacy rate (percentage
NATIONAL UNIVERSITY OF SINGAPORE
Department of Statistics and Applied Probability
2016/17 Semester 2
1.
ST3131 Regression Analysis
A regression model
data were summarized as follows.
33
289
85
Tutorial 2
was fitted to eleven observations. The
66
506
142
2
Chapter 10
Additional nodes:
Logic behind inferential methods
for comparing two groups
(Independent Samples)
Comparing Two Proportions
Comparing Two Proportions:
Confidence Interval Methods
If the following assumptions are satisfied,
Categorical response
ST 1131 Introduction to Statistics
Brief Review
From CH06 to CH12
1
Chapter 6
Probability Distributions
2
Chapter 6 Probability Distributions
Random Variable
Discrete
Continuous
Probability Distribution
Mean / the expected value
Standard deviation
Solutions to Tutorial 2 1. (a) Y = 10.2 + 4.00X (SE ) (0.6633) (0.4690) M SE = 2.199289, see code (R code) for the plot. Yes, the linear regression function ts the data well. (b) Y = 10.2 + 4.00 1 = 14.2 (c) b1 t(0.975, 8) s(b1 ) = 4 2.306 0.469 = [2.9184
Solutions to Tutorial 4 1. An output of a simple linear regression model Yi = 0 + 1 Xi + i , is as follows Coecients: Estimate Std. Error t value P-value (Intercept) -0.07727 0.12005 -0.644 0.537814 x 0.97295 0.14345 6.783 0.000140 Residual standard error
Tutorial 5 1. Suppose we have n = 10 observations (Xi , Yi ) and t the data with model
Yi = 0 + 1 Xi + i with i , i = 1, ., 10 are IID N (0, 2 ). We have the following calculations.
n
X = 0.5669,
n i=1
Y = 0.9624,
i=1 n
Yi2 = 10.2695,
Xi2
= 4.0169,
i=1
Solution to Tutorial 6 1.
For each of the following regression models, indicate whether it is a general linear regression model. If not, state whether it can be expressed in the form of a linear regression model after some suitable transformation a. b. c.
Tutorial 7 1.
A student stated: Adding predictor variables to a regression model can never reduce R2 , so we should include all available predictor variables in the model. Comment.
Bigger R2 , means the tting is better. Better tting does not imply better
Tutorial 10 1. Derive the weighted least square normal equations for tting a simple linear regression func2 tion when i = kXi , where k > 0 is a constant.
Let Qw (b0 , b1 ) =
n i=1
1 (Yi b0 b1 Xi )2 kXi
and Qw (b0 , b1 ) = 2 b0
n
Qw (b0 , b1 ) = 2 b1
i=1
Tutorial Questions 1 1. For model Yi = 0 + 1 Xi + i assume that X = 0 is within the scope of the model. What is the implication for the regression function Yi = 0 + 1 Xi if 0 = 0 so that the model is Yi = 1 Xi + i ? How would the regression function Yi =
Tutorial 2 1. Airfreight breakage A substance used in biological and medical research is shipped by airfreight to users in cartons of 1000 ampules. In the (data), X is the number of times the carton was transferred from one aircraft to another over the sh