5 Step Hypothesis for Regression
Team D will conduct a test on the hypotheses :
H₀: M₁ ≤ M₂
The null hypothesis states that non-European Union countries (M₁) have a lesser/equal to life expectancy than European Union countries (M₂). H₁: M₁ > M₂
The alternative hypothesis states that non-European Union (M₁) countries have a greater life expectancy than European Union countries (M₂). Team D will conduct research with a level of significance of α = .05 Identify the test statistic: Team D will use the results from our regression analysis to determine whether the slope of the regression line differs significantly from zero. The decision rule is with 95% confidence interval of the 63 sample observations the rejection of the null hypothesis if the computation of the p-value is greater than 2.87 and if the p-value is less than 2.87 the null hypothesis is accepted. Regression Analysis| | | | | |

...The five-step processes for hypothesis testing are the following.
Step1. Specify the null hypothesis H0 and alternative hypothesis H1. The null hypothesis is the hypothesis that the researcher formulates and proceeds to test. If the null hypothesis is rejected after the test, the hypothesis to be accepted is called the alternative hypothesis. For example if the researcher wants to compare the average value generated by two different procedures the null hypothesis to be tested is [pic] and the alternative hypothesis is [pic]
Step2. Specify the significance level ((). The significance level of a test is the probability of rejecting a null hypothesis when it is true. A test is to be constructed in such a manner that the probability of committing this error (Type I error) a pre assigned value.
Step3. Specify the test statistic and its sampling distribution. The decision to accept or reject a null hypothesis is to be based on the sample values. But, the sample values as such cannot be used for this purpose. Hence, we use a statistic, called test statistic, for this purpose. Based on the value of this test statistic we decide either to reject or accept the null hypothesis.
Step4. The fourth step is to calculate the probability value...

...Regression Analysis: A Complete Example
This section works out an example that includes all the topics we have discussed so far in this chapter.
A complete example of regression analysis.
PhotoDisc, Inc./Getty Images
A random sample of eight drivers insured with a company and having similar auto insurance policies was selected. The following table lists their driving experiences (in years) and monthly auto insurance premiums.
Driving Experience (years) Monthly Auto Insurance Premium
5 2 12 9 15 6 25 16
$64 87 50 71 44 56 42 60
a. Does the insurance premium depend on the driving experience or does the driving experience depend on the insurance premium? Do you expect a positive or a negative relationship between these two variables? b. Compute SSxx, SSyy, and SSxy. c. Find the least squares regression line by choosing appropriate dependent and independent variables based on your answer in part a. d. Interpret the meaning of the values of a and b calculated in part c. e. Plot the scatter diagram and the regression line. f. Calculate r and r2 and explain what they mean. g. Predict the monthly auto insurance premium for a driver with 10 years of driving experience. h. Compute the standard deviation of errors. i. Construct a 90% confidence interval for B. j. Test at the 5% significance level whether B is negative. k. Using α = .05, test whether ρ is different from zero.
Solution a. Based...

...diagram of number of sales calls and number of units sold
b) Estimate a simple linear regression model to explain the relationship between number of sales calls and number of units sold
y=2.139x-1.760
Number of units sold=2.139Number of units sold-1.760
c) Calculate and interpret the coefficient of correlation
r=0.853=0.9236 (There is strong correlation between two variables as its near 1)
d) the coefficient of determination
r2=0.853(The magnitude of the coefficient of determination indicates the proportion of variance in one variable, explained from knowledge of the second variable)
e) the standard error of estimate
S.E=0.3133(The standard error is the estimated standard deviation of a statistic)
f) Conduct a test of hypothesis to determine whether the coefficient of correlation in the population is zero
H0:β1=0
Ha:β1≠0
t=β1SE =6.826
p-value for df=9 and t=6.826:0.001
0.0001<0.05
Therefore null hypothesis is rejected
Hence coefficient of correlation is zero is rejected
Therefore there is significant relationship between number of sales calls and number of units sold.
g) Construct and interpret confidence intervals and prediction intervals for the dependent variable, number of units sold.
Confidence interval:
(x-tsn,x+tsn)
Confidence interval for number of sales calls:
(x-tsn,x+tsn)
(0.924, 2.8612)
CALCULATIONS ON EXCEL
Regression Analysis | | |...

...of this random factor, sample may not be exactly representative
* sampling error
* the difference between the sample mean and the population mean
* ensure that you have enough participants so that you get an accurate reflection of the population that you are interested in
* population mean (parameter), sample mean (statistic)
* the larger the samples, the closer to the population parameter the statistics will be
* probabilities
* the number possible outcomes that you are interested in divided by the total number of possible outcomes associated with an event
* null hypothesis significance testing (NHST)
* the null hypothesis states that there is no effect in the population of interest
* if the probability of obtaining the data is high, the null hypothesis is true
* no effect in the population
* distribution
* normal distribution
* skewness
* the peak of the distribution is shifted away from the middle of the graph to either left or the right
* bimodal distribution – two identifiable peaks
* often the participants drawn from two populations; you would try to identify the two different populations and analyze the data from each group separately
* parametric tests
* making assumptions about the parameters of the underlying population
* you need to ensure...

...Chapter-11
Testing of Hypothesis:
(Non-parametric Tests)
Chapter-11: Testing of Hypothesis - (Non-parametric Tests)
2
11.1. Chi - square ( χ )Test / Distribution
2
11.1.1. Meaning of Chi - square ( χ )Test
2
11.1.2. Characteristics of Chi - square ( χ )Test
2
11.2. Types of Chi - square ( χ )Test / Distribution
2
11.2.1. Chi - square ( χ )Test for Population Variance
2
11.2.2. Chi - square ( χ )Test for Goodness-of-Fit
2
11.2.3. Chi - square ( χ )Test or Independence
11.3. Analysis of Variance (ANOVA)
11.3.1. Meaning of ANOVA
11.3.2. ANOVA Approach
11.4. ANOVA Technique
11.4.1. One-way ANOVA
11.4.2. Two-way ANOVA
11.4.3. ANOVA in Latin-square Design
11.5. Other Nonparametric Techniques
Summary:
Key Terms:
Questions:
11.1. CHI-SQUARE (
) TEST /DISTRIBUTION
2
11.1.1. Meaning of Chi - square ( χ )Test
2
A chi-square test (also chi squared test or χ test) is any statistical hypothesis test in which the
sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true,
or any in which this is asymptotically true, meaning that the sampling distribution (if the null
hypothesis is true) can be made to approximate a chi-square distribution as closely as desired by
making the sample size large enough. The Chi-Square (
) test is the most popular non-parametric
test/methods, to test the hypothesis. The...

...Applied Linear Regression Notes set 1
Jamie DeCoster
Department of Psychology
University of Alabama
348 Gordon Palmer Hall
Box 870348
Tuscaloosa, AL 35487-0348
Phone: (205) 348-4431
Fax: (205) 348-8648
September 26, 2006
Textbook references refer to Cohen, Cohen, West, & Aiken’s (2003) Applied Multiple Regression/Correlation
Analysis for the Behavioral Sciences. I would like to thank Angie Maitner and Anne-Marie Leistico for
comments made on earlier versions of these notes. If you wish to cite the contents of this document, the
APA reference for them would be:
DeCoster, J. (2006). Applied Linear Regression Notes set 1. Retrieved (month, day, and year you
downloaded this ﬁle, without the parentheses) from http://www.stat-help.com/notes.html
For future versions of these notes or help with data analysis visit
http://www.stat-help.com
ALL RIGHTS TO THIS DOCUMENT ARE RESERVED
Contents
1 Introduction and Review
1
2 Bivariate Correlation and Regression
9
3 Multiple Correlation and Regression
21
4 Regression Assumptions and Basic Diagnostics
29
5 Sequential Regression, Stepwise Regression, and Analysis of IV Sets
37
6 Dealing with Nonlinear Relationships
45
7 Interactions Among Continuous IVs
51
8 Regression with Categorical IVs
59
9 Interactions involving...

...4 Hypothesis testing in the multiple regression model
Ezequiel Uriel
Universidad de Valencia
Version: 09-2013
4.1 Hypothesis testing: an overview
4.1.1 Formulation of the null hypothesis and the alternative hypothesis
4.1.2 Test statistic
4.1.3 Decision rule
4.2 Testing hypotheses using the t test
4.2.1 Test of a single parameter
4.2.2 Confidence intervals
4.2.3 Testing hypothesis about a single linear combination of the parameters
4.2.4 Economic importance versus statistical significance
4.3 Testing multiple linear restrictions using the F test.
4.3.1 Exclusion restrictions
4.3.2 Model significance
4.3.3 Testing other linear restrictions
4.3.4 Relation between F and t statistics
4.4 Testing without normality
4.5 Prediction
4.5.1 Point prediction
4.5.2 Interval prediction
4.5.3 Predicting y in a ln(y) model
4.5.4 Forecast evaluation and dynamic prediction
Exercises
1
2
2
3
55
16
17
21
21
22
26
27
28
29
30
30
30
34
34
36
4.1 Hypothesis testing: an overview
Before testing hypotheses in the multiple regression model, we are going to offer
a general overview on hypothesis testing.
Hypothesis testing allows us to carry out inferences about population parameters
using data from a sample. In order to test a hypothesis in statistics, we must perform the
following steps:
1) Formulate a...

...Topic 8: Multiple Regression Answer
a.
Scatterplot
120 Game Attendance 100 80 60 40 20 0 0 5,000 10,000 15,000 20,000 25,000 Team Win/Loss %
There appears to be a positive linear relationship between team win/loss percentage and
game attendance. There appears to be a positive linear relationship between opponent win/loss percentage and game attendance.
There appears to be a positive linear relationship between games played and game
attendance. There does not appear to be any relationship between temperature and game attendance.
b. Game Attendance Game Attendance Team Win/Loss % Opponent Win/Loss % Games Played Temperature Team Win/Loss % Opponent Win/Loss % Games Played Temperature
1 0.848748849 1 0.414250332 0.286749997 1 0.599214835 0.577958172 0.403593506 1 -0.476186226 -0.330096097 -0.446949168 -0.550083219
1
No alpha level was specified. Students will select their own. We have selected .05. Critical t = + 2.1448 t for game attendance and team win/loss % = 0.8487/ (1 − 0.84872) /(16 − 2) = 6.0043 t for game attendance and opponent win/loss % = 0.4143/ (1 − 0.41432) /(16 − 2) = 1.7032 t for game attendance and games played = 0.5992/ (1 − 0.59922) /(16 − 2) = 2.8004 t for game attendance and temperature = -0.4762/ (1 − ( − 0.4762 ) ) /(16 − 2) = -2.0263 There is a significant relationship between game attendance and team win/loss % and games played. Therefore a multiple regression model could be effective. Multiple...