LSD and Tukey yardsticks, and two-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 two-way ANOVA	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:51
LSD and Tukey yardsticks, and two-way ANOVA - SES2003	Has some problems	Chris Graham	03/12/2020 20:41
LSD and Tukey yardsticks, and two-way ANOVA - Mock - SES2003	Needs to be tested	Chris Graham	25/02/2019 11:00

Question statement

To test the effectiveness of sun-tan creams, five volunteers A, B, C, D, E each tried four creams W, X, Y, Z on various parts of their legs. They were then subjected to ultra-violet radiation and an estimate of the degree of burning was made (higher figures indicate greater burning). The results are given below with some totals:

	W	X	Y	Z	Totals
A	{r[0][0]}	{r[0][1]}	{r[0][2]}	{r[0][3]}	{t[0]}
B	{r[1][0]}	{r[1][1]}	{r[1][2]}	{r[1][3]}	{t[1]}
C	{r[2][0]}	{r[2][1]}	{r[2][2]}	{r[2][3]}	{t[2]}
D	{r[3][0]}	{r[3][1]}	{r[3][2]}	{r[3][3]}	{t[3]}
E	{r[4][0]}	{r[4][1]}	{r[4][2]}	{r[4][3]}	{t[4]}
Totals	{cols[0]}	{cols[1]}	{cols[2]}	{cols[3]}	{tot}

Test the null-hypothesis using two-way ANOVA that the creams are equally effective.

Write down the sample mean for each sun-cream together with the LSD and Tukey yardsticks so that you can see if there is any significant difference between the sample means given by these yardsticks..

Give any introductory information the student needs.

			Name	Type	Generated Value

lsd	number	7.95
tukey	number	10.84
stderror	number	2.58
bbss	number	99.00
cols	list	[ 190, 201, 162, 242 ]
vrbb	number	0.74
w6	list	[ 1, 0, 0 ]
vr	number	6.60
w4	list	[ 0, 0, 1 ]
w3	list	[ 0, 1, 0 ]
w2	list	[ 0, 0, 1 ]
w1	list	[ 0, 0, 1 ]
dfr	integer	12
rt	list	Nested 4×5 list
rs	number	33.35
t90	number	13.36
tot	number	795
sqrms	number	5.77
sig	number	6.8
tol	number	0.01
btss	number	660.55
tss	number	1159.75
dfbt	integer	3
yn	list	Nested 6×3 list
msbt	number	220.18
ssq	number	32761
v1	list	[ 0, 1, 0, 0, 0 ]
dfbb	integer	4
t99	number	20.09
t95	number	15.51
msbb	number	24.75
rss	number	400.2
me	list	[ 38, 40.2, 32.4, 48.4 ]
r1	list	Nested 4×5 list
m	integer	4
w5	list	[ 0, 1, 0 ]
n	integer	5
mu	list	[ 40, 39, 30, 43, 28 ]
r	list	Nested 5×4 list
t	list	[ 174, 156, 147, 156, 162 ]
w	list	Nested 6×3 list
v	list	[ 0, 1, 0, 0, 0 ]
pvalue	number	0.007

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(2.179*sqrms*sqrt(2/5),2)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `number`

7.95

← Depends on:

sqrms

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Choose one from a list
Part c) Choose one from a list
Part d) Choose one from a list
Part e) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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5. Gap Gap 4. Number entry
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6. Gap Gap 5. Number entry
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Part f) Match choices with answers

Gap-fill

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Here is the ANOVA table corresponding to this data:

Source	df	SS	MS	VR
Between Treatments	$\var{m-1}$	$\var{btss}$	$\var{msbt}$	$\var{vr}$
Between Blocks	$\var{n-1}$	$\var{bbss}$	$\var{msbb}$	$\var{vrbb}$
Residual	$\var{dfr}$	$\var{rss}$	$\var{rs}$	-
Total	$\var{m*n-1}$	$\var{tss}$	-	-

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

This will be used later to calculate the yardsticks.

Advice

Using the Yardsticks

The mean values for each sun-cream are:

	$\overline{x}_i$
W	$\var{me[0]}$
X	$\var{me[1]}$
Y	$\var{me[2]}$
Z	$\var{me[3]}$

The differences between the mean values for the sun-creams are:

Between $W$ and $X=\;|\var{me[0]}-\var{me[1]}|=\var{abs(me[0]-me[1])}$

Between $W$ and $Y=\;|\var{me[0]}-\var{me[2]}|=\var{abs(me[0]-me[2])}$

Between $W$ and $Z=\;|\var{me[0]}-\var{me[3]}|=\var{abs(me[0]-me[3])}$

Between $X$ and $Y=\;|\var{me[1]}-\var{me[2]}|=\var{abs(me[1]-me[2])}$

Between $X$ and $Z=\;|\var{me[1]}-\var{me[3]}|=\var{abs(me[1]-me[3])}$

Between $Y$ and $Z=\;|\var{me[2]}-\var{me[3]}|=\var{abs(me[2]-me[3])}$

We compare these differences with the LSD and Tukey yardsticks:

LSD yardstick = $2.179\times\var{sqrms}\times\sqrt{2/\var{n}}=\var{lsd}$ to 2 decimal places, where $\var{sqrms}$ is the value of $\sqrt{RMS}$ found above.

Tukey yardstick = $4.2\times\var{sqrms}\times\sqrt{1/\var{n}}=\var{tukey}$ 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 sun-creams:

Pairs of Sun-creams	Definite Significant Difference	Possible Significant Difference	No Significant Difference
Means of W and X	{yn[0][0]}	{yn[0][1]}	{yn[0][2]}
Means of W and Y	{yn[1][0]}	{yn[1][1]}	{yn[1][2]}
Means of W and Z	{yn[2][0]}	{yn[2][1]}	{yn[2][2]}
Means of X and Y	{yn[3][0]}	{yn[3][1]}	{yn[3][2]}
Means of X and Z	{yn[4][0]}	{yn[4][1]}	{yn[4][2]}
Means of Y and Z	{yn[5][0]}	{yn[5][1]}	{yn[5][2]}

Give a worked solution to the whole question.

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Newcastle University Mathematics and Statistics 9 years, 1 month ago

Error in variable testing condition

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Generated value: `number`

Parts

Gap-fill

Using the Yardsticks

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LSD and Tukey yardsticks, and two-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

Error

Warning

Generated value: number

Parts

Gap-fill

Using the Yardsticks

Generated value: `number`