Question

A study reported in the Journal of the American Medical Association investigated the cardiac demands of heavy snow shoveling. Ten healthy men underwent exercise testing with a treadmill and a cycle ergometer modified for arm cranking. The men then cleared two tracts of heavy, wet snow (ouch!) using a lightweight plastic snow shovel and an electric snow thrower. Each subject’s heart rate was recorded under the 4 exercise modes. The Excel spreadsheet provides the heart rates for ten subjects under the 4 exercise modes. Use a 5% level of significance and test if the 4 exercise modes lead to significantly different heart rates. What is your conclusion? Does the analysis indicate that there is a significant difference between the 10 subjects also? Set up the NULL and ALTERNATE hypotheses for this clearly (Note: the main goal is to study the 4 exercise modes, but you would like to study the differences between subjects as a secondary objective). Support your conclusions with statistical evidence. State any relevant test statistics clearly, with associated values and associated critical values. (and if you deal with heavy engineering products, or even lightweight products, some consumer testing such as this may be relevant to ensure safety and usability of your product).

1.What type of statistical study will you use for this experiment?

An analysis based on the central limit theorem.

A regression model of heart rate vs. exercise modes 1 and 2.

Randomized block design with 4 exercise modes as treatment.

Completely randomized design with 10 subjects as treatment.

2.Which of the following statement is true?

Group of answer choices

There is a significant difference between the 10 subjects based on the numbers 19805.2, 3 ,6601.73

There is a significant difference between exercise modes based on the numbers 22.42, 1.702E-07 and 2.96

There is a significant difference between exercise modes based on the numbers 1.05,0.42,2.25

There is a significant difference between the 10 subjects based on the numbers 1.05,0.42,2.25

3.What is the CORRELATION between the readings for the 10 subjects for the following two exercise modes:

- Correlation between the arm crank ergometer and the snow thrower readings?

	HEART RATES FOR 10 SUBJECTS USING 4 EXERCISE MODES
Subject	Treadmill	Arm-Crank Ergometer	Snow Shovel	Snow Thrower
1	177	205	180	98
2	151	177	164	120
3	184	166	167	111
4	161	152	173	122
5	192	142	179	151
6	193	172	205	158
7	164	191	156	117
8	207	170	160	123
9	177	181	175	127
10	174	154	191	109

Answer 1

count, ni =	10	10	10	10
mean , x̅ i =	178.000	171.00	175.00	123.60
std. dev., si =	16.766	18.886	14.802	18.36
sample variances, si^2 =	281.111	356.667	219.111	336.933
total sum	1780	1710	1750	1236	6476
grand mean , x̅̅ =	Σni*x̅i/Σni =	161.90

square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	259.210	82.810	171.610	1466.890
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	2592.100	828.100	1716.100	14668.900	19805.2
SS(within ) = SSW = Σ(n-1)s² =	2530.000	3210.000	1972.000	3032.400	10744.4000

no. of treatment , k =   4
df between = k-1 =    3
N = Σn =   40
df within = N-k =   36
  
mean square between groups , MSB = SSB/k-1 =    6601.7333
  
mean square within groups , MSW = SSW/N-k =    298.4556
  
F-stat = MSB/MSW =    22.1197

	SS	df	MS	F	p-value	F-critical
Between:	19805.20	3	6601.73	22.42	0.0000	2.96
Within:	10744.40	36	298.46
Total:	30549.60	39
					α =	0.05
			conclusion :	p-value<α , reject null hypothesis

1.What type of statistical study will you use for this experiment?

Randomized block design with 4 exercise modes as treatment.

2)

There is a significant difference between exercise modes based on the numbers 22.42, 1.702E-07 and 2.96

THANKS

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