MATH 045 Lecture Notes - Lecture 3: Analysis Of Covariance, Null Hypothesis, Covariate

97 views9 pages

goldsnake326

8 Apr 2020

School

VIU

Department

Adult Basic Education (ABE) Upgrading

Course

MATH 045

Professor

John Doe

For unlimited access to Class Notes, a Class+ subscription is required.

Document Summary

Imagine you compare the effectiveness of four different types of stimulant to keep you awake while revising statistics using a one-way anova. The null hypothesis would be that all four treatments have the same effect on the mean time kept awake. It is a good idea to keep the number of covariates to a minimum: a study was conducted to look at whether caffeine improves productivity at work in different conditions. The first independent variable was email, which had two levels: email access" and no email access". The second independent variable was caffeine, which also had two levels: caffeinated drink" and decaffeinated drink". Productivity was recorded at the end of the day on a scale of 0 (i may as well have stayed in bed) to 20 (wow! I got enough work done today to last all year). A sample of 30 (15 men and 15 women) participants who had never learnt to play a musical instrument before were recruited.

Related Questions

Hypothesis testing with ANOVA opinions about whether caffeine enhances test performance differ. You design a study to test the impact of drinks with different caffeine contents on students' test-taking abilities. You choose 21 students at random from your introductory psychology course to participate in your study. You randomly assign each student to one of three drinks, each with a different caffeine concentration, such that there are seven students assigned to each drink. You then give each of them a plain capsule containing the precise quantity of caffeine that would be consumed in their designated drink and have them take an arithmetic test 15 minutes later.

The students receive the following arithmetic test scores:

	Cola	Black Tea	Coffee
Caffeine Content (mg/oz)	2.9	5.9	13.4
	85	85	92	∑X² = 147,641
	86	89	87	G = 1,755
	82	82	80	N = 21
	75	75	89	k = 3
	66	88	96
	78	76	83
	87	82	92
	T₁ = 559	T₂ = 559	T₃ = 559
	SS₁ = 338.86	SS₂ = 338.86	SS₃ = 338.86
	n₁ = 7	n₂ = 7	n₃ = 7
	M₁ = 79.8571	M₂ = 79.8571	M₃ = 79.8571

1.) You plan to use an ANOVA to test the impact of drinks with different caffeine contents on students' test-taking abilities. What is the null hypothesis?

(a) The population mean test score for the cola population is different from the population mean test score for the black tea population.

(b) The population mean test scores for all three treatments are equal.

(d) The population mean test scores for all three treatments are not all equal.

2.) Calculate the degrees of freedom and the variances for the following ANOVA table:

Source	SS	df	MS
Between	-	-	-
Within	702.28	-	-
Total	973.14	-

The formula for the F-ratio is:

3.) Using words, the formula of the F-ratio can be written as:

4.) Using the data from the ANOVA table given, the F-ratio can be written as:

5.) Calculate for F-ratio:

6.) At the level of significance, what is your conclusion?

(a)You can reject the null hypothesis; you do not have enough evidence to say that caffeine affects test performance.

(b)You cannot reject the null hypothesis; caffeine does appear to affect test performance.

(c)You cannot reject the null hypothesis; you do not have enough evidence to say that caffeine affects test performance.

(d)You can reject the null hypothesis; caffeine does appear to affect test performance.

ceruleanfly137

QUESTION 1What is the purpose of post hoc tests? Under what circumstances are post hoc tests not needed or appropriate?

a. Under what circumstances is ANOVA a more appropriate statistical technique to use than a t test for independent samples? Please provide an example(s) to support your answer. (2 marks)

b. (2 marks)

c. In testing the difference between group means, how are between group and within group variability related to each other? Provide an example to support your answer. (2 marks)

QUESTION 2

a. If a two-factor ANOVA results in a significant main effect for factor A and a significant AxB interaction, explain why you should be cautious in interpreting the effect of factor A. (1 mark)

b. For the following research situations what would be your main effects and interaction effect. If there was a significant interaction effect what would that mean? (please be as clear as possible in your explanation):

A researcher hypothesizes that the effect of alcohol consumption on one’s motor skills depends on the person’s weight (underweight, normal, overweight), but that this may differ for men versus women. (2 marks)
A teacher is interested in seeing how attending class and reading the assigned chapters is related to her students’ performance on tests. She assesses each student in terms of how often they attend class (rarely, sometimes, regularly) and how often they read the assigned chapters (rarely, sometimes, regularly). (2 marks)

QUESTION 3

Ethel is interested in two methods of note-taking strategies and the effect of these strategies on the overall GPAs of college freshmen. She believes that men would benefit most from Method 1, while women would benefit most from Method 2. After obtaining 30 men and 30 women volunteers in freshmen orientation, she randomly assigns 10 men and 10 women to Method 1, 10 men and 10 women to Method 2, and 10 men and 10 women to a control condition. During the first month of the spring semester individuals in the two note-taking method groups receive daily instruction on the particular note-taking method to which they were assigned. The control group receives no note-taking instruction. Fall and spring GPAs for all participants are recorded and a score that is the difference between the spring and fall semester GPAs is calculated.

Below are the results from running a two-way ANOVA. Please answer the following:

What is the dependent variable? (0.5 marks)
What are the independent variables (factors)? (0.5 marks)
What is the research question(s) being addressed? (1 mark)
What does the descriptive statistics (graphical) tell us about the effects under study?

(1 mark)
What assumptions are being tested in the R output? Are these assumptions met? Please

provide statistical evidence. (2 marks)
What effects are being tested in the R output? (1 mark)
What does output from the ANOVA table tell us? (1 mark)
What type of follow-up analyses was conducted and why? Please interpret. (2 marks)

: Men
: Method1

      median
coef.var

0.300

0.469

: Women
: Method1

      median
coef.var

0.150

0.496

: Men
: Method2

      median
coef.var

0.275

0.027

: Women
: Method2

      median
coef.var

0.600

0.629

: Men
: Control

      median
coef.var

0.175

mean 0.335

mean 0.170

mean 0.305

mean 0.640

mean 0.165

mean 0.105

skew.2SE
   0.408

SE.mean CI.mean.0.95

var 0.052

var 0.033

var 0.037

var 0.032

var 0.022

var 0.021

normtest.W
     0.936

std.dev
  0.229

0.682
    skewness

1.076
    skewness

0.630
    skewness

0.072

SE.mean CI.mean.0.95

normtest.p 0.830 -----------------------------------------------------------------------------

skew.2SE
   0.342

kurtosis
  -0.645

kurt.2SE
  -0.242

normtest.W
     0.964

0.058

SE.mean CI.mean.0.95

std.dev
  0.183

normtest.p 0.096 -----------------------------------------------------------------------------

skew.2SE
   0.361

kurtosis
  -1.363

kurt.2SE
  -0.511

normtest.W
     0.869

0.061

SE.mean CI.mean.0.95

std.dev
  0.192

normtest.p 0.857 -----------------------------------------------------------------------------

0.278
    skewness

0.904
    skewness

skew.2SE
  -0.124

kurtosis
  -1.096

kurt.2SE
  -0.411

normtest.W
     0.979

-0.171

: Women
: Control

      median
coef.var

0.100

1.392
    skewness

0.560

0.046

kurtosis
  -0.759

0.105

kurt.2SE
  -0.284

   std.dev
     0.146

normtest.p
     0.512

skew.2SE
   0.019

kurtosis
  -1.443

kurt.2SE
  -0.541

normtest.W
     0.967

skew.2SE
   0.458

kurtosis
  -0.770

kurt.2SE
  -0.289

normtest.W
     0.906

0.056

SE.mean CI.mean.0.95

std.dev
  0.178

normtest.p 0.254 -----------------------------------------------------------------------------

                                                                      std.dev
                                                                        0.149

normtest.p 0.958 -----------------------------------------------------------------------------

0.047

SE.mean CI.mean.0.95

0.164

0.131

0.137

0.127

0.107

> leveneTest(gpa$gpaimpr~gpa$gender, data=gpa,center=mean)

Levene's Test for Homogeneity of Variance (center = mean) Df F value Pr(>F)

group 1 7.09 0.01* 58

---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > leveneTest(gpa$gpaimpr~method, data=gpa,center=mean) Levene's Test for Homogeneity of Variance (center = mean)

      Df F value Pr(>F)
group  2    1.94   0.15

> #running Levene's test for assumption of HOV (INTERACTION EFFECT)
> leveneTest(gpa$gpaimpr, interaction(gpa$gender,gpa$method),center=mean) Levene's Test for Homogeneity of Variance (center = mean)

      Df F value Pr(>F)
group  5    0.57   0.72

Df Sum Sq Mean Sq F value Pr(>F) gender 1 0.020 0.020 0.61 0.43753

method 2 1.174 0.587 17.81 1.1e-06 *** gender:method 2 0.695 0.348 10.54 0.00014 *** Residuals 54 1.780 0.033
---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

gender        0.0055
method        0.3200
gender:method 0.1894

0.011
0.397
0.281

eta.sq eta.sq.part

#selecting only Method1, run ANOVA, then follow-up tests

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = gpaimpr ~ gender, data = gpaMethod1)
$`gender`
Women-Men -0.165 -0.3594851  0.02948506    0.0915581

diff        lwr        upr     p adj

#selecting only Method2, run ANOVA, then follow-up tests

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = gpaimpr ~ gender, data = gpaMethod2)
$`gender`
Women-Men 0.335 0.161153 0.508847   0.000754

diff      lwr      upr    p adj

aquamarineox434

You and your hypothetical lab partner each develop an experimental plan to test this hypothesis. The two experimental plans are presented below. Evaluate each plan for its pros and cons in the experimental design using the bullet points below. Both experimental plans have some good elements and areas that could be improved.

Experiment A) Mature 2 inch sections (sprigs) of Elodea will be used to measure oxygen production under different light conditions. 5 sprigs will be tested. One will be kept in the dark (no light). The others will be exposed to a light filtered with cellophane. One sprigâs light will be filtered through clear cellophane to receive all wavelengths of light, and the other sprigsâ light will be filtered through red, blue, and green cellophane, respectively. The no-light plant will be kept in a dark cabinet, while the other plants will be together on a counter. Each plant sprig will be placed in equal sized upside-down test tubes in 100ml of tap water so that oxygen bubbles produced by the plant will be trapped at the top of the test tube. A ruler will be used to measure the length of the oxygen pocket collected at the top of the test tube. The experiment will continue until one sprig produces 5 cm of oxygen in the tube.

Experiment B) 3 cm sections (sprigs) of Elodea of various ages will be used to measure oxygen production under different light conditions. 15 sprigs will be divided into 3 groups of 5 sprigs each. One group will receive natural sunlight, one group will be exposed to a blue light bulb under a box to block natural light, and the final group will be exposed to a red light bulb under a similar box to block natural light. Each group of plant sprigs will be trapped under a clear plastic funnel that is covered by an upside-down test tube to collect oxygen bubbles produced by all the sprigs in the group. All groups will be in 100ml of water of the same temperature and pH. The experiment will continue for 24 hours, and then the volume of oxygen gas produced will be measured by multiplying the length of the oxygen pocket by the circle described by the test tube, ?r². The average volume produced by all groups will be graphed and compared to determine which condition produced the most total volume of oxygen.

Consider the following questions to evaluate experimental plans A and B (DO NOT ANSWER THESE QESTIONS)

What is the independent variable suggested by the hypothesis? Do both plans test the independent variable equally well?

What is the dependent variable in each experiment? Which experimentâs dependent variable best matches the prediction in the hypothesis?

What are the experimental and control groups for each experiment? Which experiment gives the best opportunity to assess the effect of the independent variable?

How does each experiment account for standardized variables? Which experiment do you think is the better controlled experiment?

How is sample size addressed in each experiment? Which would you find to be more convincing?

Does either experiment describe any statistical analyses to be performed on the data?

ANSWER THIS QUESTION: BELOW

Considering your analysis of each of the questions above,write an improved experimental design to test the hypothesis âIf blue light is more effective at promoting photosynthesis, then more oxygen gas (O₂) will be produced when plants are exposed to blue-filtered light compared to other wavelengths.âYou may use elements of the above experiments

lilaccod491

MATH 045 Lecture Notes - Lecture 3: Analysis Of Covariance, Null Hypothesis, Covariate

Document Summary

Get access

Related textbook solutions

Chemistry: Structure and Properties

Basic Chemistry

Principles of Chemistry Molecular Approach

Calculus

Principles of Chemistry Molecular Approach

Chemistry: Structure and Properties

Single Variable Calculus: Early Transcendentals

Chemistry: A Molecular Approach

Chemistry: A Molecular Approach

CALCULUS:EARLY TRANSCENDENTALS

Principles of Chemistry: A Molecular Approach

Chemistry: The Central Science

Related Documents

PSYC 333 Lecture Notes - Lecture 13: Observational Error, Total Variation, Dependent And Independent Variables

PSYC 316 Lecture Notes - Lecture 10: Libido, Sildenafil, Type I And Type Ii Errors

PSYC 305 Lecture Notes - Lecture 21: Total Variation, Analysis Of Covariance, Dependent And Independent Variables

Related Questions