Question

a physical trainer has four workouts that he recommends for his clients. the workouts have been designed so that the average maximum heart rate achieved is the same for each workout. to test this design he randomly selects twenty people and randomly assigns five of them to use each of the workouts. during each workout, he measures the maximum heart rate in beats per minute with the following results. can the physical trainer conclude that there is a difference among the average maximum heart rates which are achieved during the four workouts?

maximum heart rates (beats per minute)
151	187	185	173
154	194	186	177
156	156	179	161
156	176	190	161
177	163	167	156

step 1 of 2: find the value of the test statistic to test for a difference between the workouts. round your answer to two decimal places, if necessary.

answer

f =

Explanation:

Step1: Calculate group means and grand mean

Let $x_{ij}$ be the $j$-th observation in the $i$-th group.
For Workout #1: $\bar{x}_1=\frac{151 + 154+156+156+177}{5}=\frac{794}{5} = 158.8$
For Workout #2: $\bar{x}_2=\frac{187+194 + 156+176+163}{5}=\frac{876}{5}=175.2$
For Workout #3: $\bar{x}_3=\frac{185+186+179+190+167}{5}=\frac{907}{5}=181.4$
For Workout #4: $\bar{x}_4=\frac{173+177+161+161+156}{5}=\frac{828}{5}=165.6$

The grand - mean $\bar{\bar{x}}=\frac{794 + 876+907+828}{20}=\frac{3405}{20}=170.25$

Step2: Calculate sum of squares between groups (SSB)

$SSB=\sum_{i = 1}^{k}n_i(\bar{x}_i-\bar{\bar{x}})^2$, where $k = 4$ (number of groups) and $n_i=5$ (number of observations in each group)
$SSB=5\times[(158.8 - 170.25)^2+(175.2-170.25)^2+(181.4 - 170.25)^2+(165.6-170.25)^2]$
$SSB=5\times[(- 11.45)^2+(4.95)^2+(11.15)^2+(-4.65)^2]$
$SSB=5\times(131.1025 + 24.5025+124.3225 + 21.6225)$
$SSB=5\times301.55$
$SSB = 1507.75$

Step3: Calculate sum of squares within groups (SSW)

$SSW=\sum_{i = 1}^{k}\sum_{j = 1}^{n_i}(x_{ij}-\bar{x}_i)^2$
For Workout #1:
$(151 - 158.8)^2+(154 - 158.8)^2+(156 - 158.8)^2+(156 - 158.8)^2+(177 - 158.8)^2$
$=(-7.8)^2+(-4.8)^2+(-2.8)^2+(-2.8)^2+(18.2)^2$
$=60.84+23.04 + 7.84+7.84+331.24=430.8$
For Workout #2:
$(187 - 175.2)^2+(194 - 175.2)^2+(156 - 175.2)^2+(176 - 175.2)^2+(163 - 175.2)^2$
$=(11.8)^2+(18.8)^2+(-19.2)^2+(0.8)^2+(-12.2)^2$
$=139.24+353.44+368.64+0.64+148.84 = 1010.8$
For Workout #3:
$(185 - 181.4)^2+(186 - 181.4)^2+(179 - 181.4)^2+(190 - 181.4)^2+(167 - 181.4)^2$
$=(3.6)^2+(4.6)^2+(-2.4)^2+(8.6)^2+(-14.4)^2$
$=12.96+21.16+5.76+73.96+207.36 = 321.2$
For Workout #4:
$(173 - 165.6)^2+(177 - 165.6)^2+(161 - 165.6)^2+(161 - 165.6)^2+(156 - 165.6)^2$
$=(7.4)^2+(11.4)^2+(-4.6)^2+(-4.6)^2+(-9.6)^2$
$=54.76+129.96+21.16+21.16+92.16 = 319.2$

$SSW=430.8+1010.8+321.2+319.2=2082$

Step4: Calculate degrees of freedom

Degrees of freedom between groups $df_{B}=k - 1=4 - 1=3$
Degrees of freedom within groups $df_{W}=n - k=20 - 4=16$

Step5: Calculate mean square between groups (MSB) and mean square within groups (MSW)

$MSB=\frac{SSB}{df_{B}}=\frac{1507.75}{3}\approx502.58$
$MSW=\frac{SSW}{df_{W}}=\frac{2082}{16}=130.125$

Step6: Calculate the F - statistic

$F=\frac{MSB}{MSW}=\frac{502.58}{130.125}\approx3.86$

Answer:

$F = 3.86$