LSD and Tukey yardsticks, and one-way ANOVA

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 1 month ago

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Name	Status	Author	Last Modified
LSD and Tukey yardsticks, and one-way ANOVA	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:51
LSD and Tukey yardsticks, and one-way ANOVA - SES2003	Has some problems	Chris Graham	23/11/2022 13:16
LSD and Tukey yardsticks, and one-way ANOVA - Mock Test - SES2003	Needs to be tested	Chris Graham	04/03/2019 10:28

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Question statement

The following data arose in a comparison of the effects of alcohol on the time taken to complete a task. There were three groups of subjects; Group A had no alcohol, Group B had two units over 1 hour and Group C had 4 units over 1 hour. The responses are the times (in seconds) taken to complete a word-matching task.

Group A (0 units)	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$
Group B (2 units)	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$
Group C (4 units)	$\var{r3[0]}$	$\var{r3[1]}$	$\var{r3[2]}$	$\var{r3[3]}$	$\var{r3[4]}$	$\var{r3[5]}$

Give any introductory information the student needs.

			Name	Type	Generated Value

lsd	number	5.20
tukey	number	6.34
sd1	number	3.88
sd2	number	3.76
sd3	number	4.94
vr	number	9.25
w3	list	[ 1, 0, 0 ]
w2	list	[ 1, 0, 0 ]
w1	list	[ 0, 0, 1 ]
mbt	number	165.39
n	integer	18
pmsg	list	List of 5 items
mu3	number	32.5
stderror	number	7.30
sqrms	number	4.23
dfrs	integer	15
m1	number	25.67
m3	number	35.00
m2	number	26.17
btss	number	330.78
stovern	number	15410.8333333333
tss	number	598.94
dfbt	integer	2
tol	number	0.01
yn	list	Nested 3×3 list
ssq	list	[ 4028, 4179, 7472 ]
sig1	number	3.2
v1	list	[ 0, 1, 0, 0, 0 ]
sig3	number	3.2
sig2	number	3.2
cmsg	list	List of 5 items
rss	number	268.17
tsqovern	list	List of 3 items
mu1	number	23.5
g	number	521
r2	list	[ 22, 27, 24, 30, 23, 31 ]
mu2	number	28.5
ss	number	15679
k	integer	3
r3	list	[ 31, 34, 37, 44, 33, 31 ]
mrs	number	17.88
r1	list	[ 26, 26, 20, 24, 26, 32 ]
u	integer	1
t	list	[ 154, 157, 210 ]
w	list	Nested 3×3 list
v	list	[ 0, 1, 0, 0, 0 ]
n1	integer	6
n2	integer	6
n3	integer	6
pvalue	number	0.002

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(2.131*sqrms*sqrt(2/n1),2)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

5.20

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

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3. Gap Gap 2. Match choices with answers
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Ask the student a question, and give any hints about how they should answer this part.

You are given the following ANOVA table for this data:

Source	df	SS	MS	VR
Between Treatments	$\var{dfbt}$	$\var{btss}$	$\var{mbt}$	$\var{vr}$
Residual	$\var{dfrs}$	$\var{rss}$	$\var{mrs}$	-
Total	$\var{n-1}$	$\var{tss}$	-	-

Input $\sqrt{RMS}$ here: Unnamed gap to 2 decimal places.

This will be used to calculate the LSD and Tukey yardstick values later.

Advice

Using the Yardsticks

The mean values for each group are:

	$\overline{x}_i$
Group A	$\var{m1}$
Group B	$\var{m2}$
Group C	$\var{m3}$

The differences between the mean values for the groups are:

Between $A$ and $B=\;|\var{m1}-\var{m2}|=\var{abs(m1-m2)}$

Between $B$ and $C=\;|\var{m2}-\var{m3}|=\var{abs(m2-m3)}$

Between $A$ and $C=\;|\var{m1}-\var{m3}|=\var{abs(m1-m3)}$

We compare these differences with the LSD and Tukey yardsticks:

LSD yardstick = $2.131\times\var{sqrms}\times\sqrt{2/\var{n1}}=\var{lsd}$ to 2 decimal places, where $\var{sqrms}$ is the value of $\sqrt{RMS}$ found above.

Tukey yardstick = $3.67\times\var{sqrms}\times\sqrt{1/\var{n1}}=\var{tukey}$ to 2 decimal places.

If the difference of the means:

is greater than the Tukey yardstick we say that there is evidence of a definite significant difference.

less than the Tukey yardstick but greater than the LSD yardstick we say that there is evidence of a possible significant difference.

is less than the LSD yardstick then we say that there is no evidence of a significant difference.

Hence we have the following for the groups:

Pairs of Groups	Definite Significant Difference	Possible Significant Difference	No Significant Difference
Means of Groups A and B	{yn[0][0]}	{yn[0][1]}	{yn[0][2]}
Means of Groups B and C	{yn[1][0]}	{yn[1][1]}	{yn[1][2]}
Means of Groups A and C	{yn[2][0]}	{yn[2][1]}	{yn[2][2]}

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This question is used in the following exams:

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Newcastle University Mathematics and Statistics 9 years, 1 month ago

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LSD and Tukey yardsticks, and one-way ANOVA

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`