Question

Forty-minute workouts of one of the following activities three days a week will lead to a loss of weight. Suppose the following sample data show the number of calories burned during 40-minute workouts for three different activities.

Swimming	Tennis	Cycling
403	410	385
375	490	250
430	450	290
400	415	397
432	535	263

Do these data indicate differences in the amount of calories burned for the three activities? Use a 0.05 level of significance.

Find the value of the test statistic.

Find the p-value. (Round your answer to three decimal places.)

p-value =

Answer 1

Null hypotheses H0 : The average amount of calories burned for the three activities are equal.

Alternative hypotheses H1 : At least one of the three activities have different amount of calories burned.

Let Ti be the total calories burned for group i, ni be number of observations of group i.

Let G be the total calories burned of all observations and N be total number of observations.

Σ is sum of squares of calories burned of all observations

T1 = 2040, T2 = 2300 , T3 = 1585

G = 2040 + 2300 + 1585 = 5925

Σ = 834558 + 1069150 + 521603 = 2425311

SST = Σ - /N = = 84936

SSTR = Σ/n - /N = = 52390

SSE = 84936 - 52390 = 32546

Degree of freedom of group = Number of level - 1 = 3 - 1 = 2

Degree of freedom of error = Number of observations - Number of level = 15 - 3 = 12

MSTR = SSTR / DF group = 52390 / 2 = 26195

MSE = SSE / DF error = 32546 / 12 = 2712.167

Test Statistic, F = MSTR / MSE = 26195 / 2712.167 =  9.658

Test statistic will follow F distribution with degree of freedom 2 , 12.

P-value = P(F > 9.658) =  0.003

Since, p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is strong evidence from the data that at least one of the three activities have different amount of calories burned.

R Output -

> # Fit a regression model on scale for different factors of Lake and run the ano .... [TRUNCATED]
Df Sum Sq Mean Sq F value Pr(>F)   
calories 2 52390 26195 9.658 0.00317 **
Residuals 12 32546 2712   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1