Question

A runner was tested on a treadmill. During the test, his speed x (in km/h) and his heart rate y were measured. The results are shown in the table.

y	126	137	149	163	180	193
x	7	9	11	13	15	17

(a) Test for the significance of regression using the analysis of variance with α=0.01. Find the P-value for this test. Can you conclude that the model specifies a useful linear relationship between these two variables?

We Choose your answer in accordance to the item a) of the question statement

cannotcan

conclude that the model specifies a useful linear relationship at α=0.01.

(b) Estimate σ2.
Round your answer to three decimal places (e.g. 98.765).

σ^2= Enter your answer in accordance to the item b) of the question statement

(c) Estimate the standard error of the slope and intercept in this model.
Round your answers to three decimal places (e.g. 98.765).

se(β^1)= Enter your answer; se(beta1)
se(β^0)= Enter your answer; se(beta0)

(d) Test the hypothesis that the increase in the speed of 1 km/h results in the runner’s heart rate average increase of 7 points at α=0.01. Suppose that the alternative hypothesis is that the average increase of the runner’s heart rate in this situation does not equal 7 points.

There is Choose your answer in accordance to the item d) of the question statement

enoughnot enough

evidence to conclude that the increase in the speed of 1 km/h does not result in the runner’s heart rate average increase of 7 points at α=0.01.

Answer 1

Anova table
variation	SS	df	MS	F-stat	p-value
regression	3264.06	1	3264.0571	818.9391	0.0000
error,	15.94	4	3.9857
total	3280.000	5

a)

F stat = 818.9391

p value=0.000

decison :    p-value<α , reject Ho
reject Ho and conclude  that linear relations exists between X and y  

b)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	72.00	948.00	70.00	3280.00	478.00
mean	12.00	158.00	SSxx	SSyy	SSxy

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    15.942857
estimate of variance,   Se² = SSE/(n-2) =    3.986

c)

estimated std error of slope =Se(ß1) = Se/√Sxx =    1.9964/√70=   0.239

estimated std error of intercept =Se(ßo) =    Se*√(1/n+x̅²/Sxx)=       2.977

d)

slope hypothesis test      
Ho:   β1=   7
H1:   β1╪   7
n=   6  
alpha =   0.01  
estimated std error of slope =Se(ß1) = Se/√Sxx =    1.9964/√70=   0.2386
t stat = estimated slope/std error =ß1 /Se(ß1) =    (6.8286-7)/0.2386=   -0.72
Degree of freedom ,df = n-2=   4  
t-critical value=    4.6041   [excel function: =T.INV.2T(α,df) ]
      
p-value =    0.5122  
decison :    p-value>α , do not reject Ho  

not enough

There is not enough evidence to conclude that the increase in the speed of 1 km/h does not result in the runner’s heart rate average increase of 7 points at α=0.01.