Which one of the following is a correct pair of hypotheses for the F-test? H0: BrainChange is the same for all three activitiesH1: BrainChange is different for all three activitiesH0: The underlying mean BrainChange is the same for all three activitiesH1: The underlying mean BrainChange is different for at least one activityH0: BrainChange is different for all three activitiesH1: BrainChange is the same for all three activitiesH0: The underlying mean BrainChange is the same for all three activitiesH1: The underlying mean BrainChange is different for all three activitiesH0: The underlying mean BrainChange is different for all three activitiesH1: The underlying mean BrainChange is the same for all three activities. Table 1: One-sample t-test outputQuestions 10 to 14 refer to the following additional information.A one-way analysis of variance (ANOVA) F-test was conducted to investigate the relationship between Activityand BrainChange.LetTai be the underlying mean BrainChangewhen the activity is TaiChi"Soc be the underlying mean BrainChangewhen the activity is Socialand recall that"NoInis the underlying mean BrainChangewhen there is no activity assigned.The plot of BrainChange by Activity is shown in Figure 2 (with each group mean being indicated by atriangle) and the output from the F-test is shown in Table 2.TaiChiSocialNoIntervention00-2BrainChangeTaiChiSocialNolnterventionTotal00BrainChangeFigure 2: BrainChange by ActivityBetween GroupsWithin GroupsTotal0O OOD OSocial-1Sum ofSquares7.82069.55677.37600 00NMean29 4711.8557527 4056 .6969024 -.2401 1.2584380 2356 98967Dependent Variable: BrainChangeTukey HSD(1) Treatment (J) TreatmentTaiChiSocialNolnterventionTaiChiNolntervention TaiChiSocial00NolnterventionNote that one of the values has been replaced with ++DescriptivesStd.Deviation Std. ErrorANOVAOdf2777900.158911341225688.110650BrainChangeMeanSquare3.910.9038 000 00⁰ 00OOO OOMean Difference(1-J).06544.71119-.06544.64575-.71119-.64575Minimum-1.83-1.36-2.03-2.03Multiple ComparisonsF++Std.Error.25418.26227.25418.26664.26227.26664Maximum2.201.802.012.20Sig..017Sig.964.022.964.046.022.0460000O0 0 0O.0844-.6729.0085-1.3380-1.28300295% Confidence IntervalLower Bound Upper Bound-.5420.67291.3380.54201.2830-.0844-.0085 Brain size typically shrinks as people age and such shrinkage may be linked to dementia. A study was done inChina to investigate whether different types of activity might help to prevent brain shrinkage or perhaps even leadto an increase in brain size.The 80 (non-demented) volunteers were aged between 60 and 69. Each of them had an MRI to determine brainsize before the study began. They were then randomly allocated to one of three activity groups: TaiChi, Social orNolntervention.Except for the Nolntervention group, the groups met with their leaders three times a week over a period of 40weeks. The Nolntervention group were called four times by the study coordinator during the study period.At the end of the study, the participants had another MRI and the percentage change in size of each participant'sbrain was calculated.Adapted from: Mortimer, J. A., Ding, D., Borenstein, A. R., DeCarli C., Guo, Q., Wu, Y., Zhao, Q. and Chu, S.(2012), Changes in Brain Volume and Cognition in a Randomized Trial of Exercise and Social Interaction in aCommunity-Based Sample of Non-Demented Chinese Elders.The two variables used are defined as:ActivityThe activity the participant was randomly assigned to- TaiChi(meet two Tai Chi masters to practise Tai Chi)- SocialBrainChange Percentage change in size of participant's brain(meet at a community centre to discuss interesting topics)- Nolntervention(no activity assigned)Questions 8 and 9 refer to the following additional information.(where a positive value means the brain increased in size)A plot of BrainChange for the participants who were assigned to the NoIntervention group is shown in Figure 1.The mean is indicated by the triangle.NolnterventionooooooNoIntervention0BralnChangeFigure 1: BrainChange for the Nolntervention group-100One-Sample StatisticsN24Let μNoInt be the underlying mean BrainChange when there is no activity assigned.A one-sample t-test with hypotheses Ho: "NoInt = 0 vs H₁ : NoInt #0 was conducted to investigate whetherthere was an underlying mean BrainChange when there was no intervention. The output for this test is shown inTable 1.-.935O00 00df23Mean Std. Deviation Std. Error Mean-.24011.2584325688One-Sample TestTest Value = 0OSig. (2-tailed) Mean Difference.360-24012295% Confidence Interval of theLower-.7715DifferenceUpper2913

It is needed to find the correct pair of hypotheses for F test.

Answered: Which one of the following is a correct…

MATLAB: An Introduction with Applications

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Which one of the following is a correct pair of hypotheses for the F-test?

H₀: BrainChange is the same for all three activities

H₁: BrainChange is different for all three activities
H₀: The underlying mean BrainChange is the same for all three activities

H₁: The underlying mean BrainChange is different for at least one activity
H₀: BrainChange is different for all three activities

H₁: BrainChange is the same for all three activities
H₀: The underlying mean BrainChange is the same for all three activities

H₁: The underlying mean BrainChange is different for all three activities
H₀: The underlying mean BrainChange is different for all three activities

H₁: The underlying mean BrainChange is the same for all three activities.

Table 1: One-sample t-test output
Questions 10 to 14 refer to the following additional information.
A one-way analysis of variance (ANOVA) F-test was conducted to investigate the relationship between Activity
and BrainChange.
Let
Tai be the underlying mean BrainChange
when the activity is TaiChi
"Soc be the underlying mean BrainChange
when the activity is Social
and recall that
"NoInis the underlying mean BrainChange
when there is no activity assigned.
The plot of BrainChange by Activity is shown in Figure 2 (with each group mean being indicated by a
triangle) and the output from the F-test is shown in Table 2.
TaiChi
Social
NoIntervention
00
-2
BrainChange
TaiChi
Social
Nolntervention
Total
00
BrainChange
Figure 2: BrainChange by Activity
Between Groups
Within Groups
Total
0
O OOD O
Social
-1
Sum of
Squares
7.820
69.556
77.376
00 00
N
Mean
29 4711
.85575
27 4056 .69690
24 -.2401 1.25843
80 2356 98967
Dependent Variable: BrainChange
Tukey HSD
(1) Treatment (J) Treatment
TaiChi
Social
Nolntervention
TaiChi
Nolntervention TaiChi
Social
00
Nolntervention
Note that one of the values has been replaced with ++
Descriptives
Std.
Deviation Std. Error
ANOVA
O
df
2
77
79
00
.15891
13412
25688
.11065
0
BrainChange
Mean
Square
3.910
.903
8 000 00⁰ 00
OOO OO
Mean Difference
(1-J)
.06544
.71119
-.06544
.64575
-.71119
-.64575
Minimum
-1.83
-1.36
-2.03
-2.03
Multiple Comparisons
F
++
Std.
Error
.25418
.26227
.25418
.26664
.26227
.26664
Maximum
2.20
1.80
2.01
2.20
Sig.
.017
Sig.
964
.022
.964
.046
.022
.046
00
00
O
0 0 0
O
.0844
-.6729
.0085
-1.3380
-1.2830
0
2
95% Confidence Interval
Lower Bound Upper Bound
-.5420
.6729
1.3380
.5420
1.2830
-.0844
-.0085

Brain size typically shrinks as people age and such shrinkage may be linked to dementia. A study was done in
China to investigate whether different types of activity might help to prevent brain shrinkage or perhaps even lead
to an increase in brain size.
The 80 (non-demented) volunteers were aged between 60 and 69. Each of them had an MRI to determine brain
size before the study began. They were then randomly allocated to one of three activity groups: TaiChi, Social or
Nolntervention.
Except for the Nolntervention group, the groups met with their leaders three times a week over a period of 40
weeks. The Nolntervention group were called four times by the study coordinator during the study period.
At the end of the study, the participants had another MRI and the percentage change in size of each participant's
brain was calculated.
Adapted from: Mortimer, J. A., Ding, D., Borenstein, A. R., DeCarli C., Guo, Q., Wu, Y., Zhao, Q. and Chu, S.
(2012), Changes in Brain Volume and Cognition in a Randomized Trial of Exercise and Social Interaction in a
Community-Based Sample of Non-Demented Chinese Elders.
The two variables used are defined as:
Activity
The activity the participant was randomly assigned to
- TaiChi
(meet two Tai Chi masters to practise Tai Chi)
- Social
BrainChange Percentage change in size of participant's brain
(meet at a community centre to discuss interesting topics)
- Nolntervention
(no activity assigned)
Questions 8 and 9 refer to the following additional information.
(where a positive value means the brain increased in size)
A plot of BrainChange for the participants who were assigned to the NoIntervention group is shown in Figure 1.
The mean is indicated by the triangle.
Nolntervention
oooooo
NoIntervention
0
BralnChange
Figure 1: BrainChange for the Nolntervention group
-1
00
One-Sample Statistics
N
24
Let μNoInt be the underlying mean BrainChange when there is no activity assigned.
A one-sample t-test with hypotheses Ho: "NoInt = 0 vs H₁ : NoInt #0 was conducted to investigate whether
there was an underlying mean BrainChange when there was no intervention. The output for this test is shown in
Table 1.
-.935
O
00 00
df
23
Mean Std. Deviation Std. Error Mean
-.2401
1.25843
25688
One-Sample Test
Test Value = 0
O
Sig. (2-tailed) Mean Difference
.360
-24012
2
95% Confidence Interval of the
Lower
-.7715
Difference
Upper
2913

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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