Question

1. (30 pts) The following data are the rates of oxygen consumption of birds, measured at different
environmental temperatures:
T (°C) -18 -15 -10 -5 0 5 10 19
O2 consumption
(ml/g/hr) 5.2 4.7 4.5 3.6 3.4 3.1 2.7 1.8
a) Calculate the intercept, a, and slope, b, for the regression of oxygen consumption rate on
temperature.
b) Test the null hypothesis H0: β = 0, by analysis of variance.
c) Test the null hypothesis H0: β = 0, by t-test.
d) Calculate the standard error of the estimated slope, as well as 95% confidence interval.
e) Calculate the coefficient of determination of the regression.
(Note: Please show detailed calculation steps, and do not use computer software for this question)

2. (20 pts) The maximum heart rates (bpm) of healthy active females were recorded based on a stress
test. The subjects exercised on a tread mill for as long as they were able; this duration (in seconds) was
recorded. Is their maximum heart rate significantly determined by the amount of time they were able to
spend on the tread mill? [Adapted from Bruce et al., 1973]
(Note: This question can be done using computer software, please attach your code and print out the
results if you are using Matlab, or print/attach the Excel sheet if you are using Excel).

Duration (s)	Heart Rate (bpm)
660	184
628	183
637	200
575	170
590	188
600	190
562	190
495	180
540	184
470	162
408	188
387	170
564	184
603	175
420	180
573	200
602	190
430	170
508	158
565	186
464	166
495	170
461	188
540	190
588	194
498	190
483	190
554	188
521	184
436	170
398	168
366	175
439	156
549	184
360	180
566	184
407	156
602	194
488	190
526	188
524	164
562	185
496	178

Answer 1

1)

a)

The regression equation is defined as,

The least square estimate of intercept and slope are,

From the data values,

	X	Y	X^2	Y^2	XY
	-18	5.2	324	27.04	-93.6
	-15	4.7	225	22.09	-70.5
	-10	4.5	100	20.25	-45
	-5	3.6	25	12.96	-18
	0	3.4	0	11.56	0
	5	3.1	25	9.61	15.5
	10	2.7	100	7.29	27
	19	1.8	361	3.24	34.2
Sum	-14	29	1160	114.04	-150.4

The regression equation is,

b)

The ANOVA table is obtained in following steps,

The degree of freedom values are,

The sum of square values are,

From the data values,

X	Y
-18	5.2	5.0511	3.6250	1.4261	2.0337
-15	4.7	4.7878	3.6250	1.1628	1.3521
-10	4.5	4.3490	3.6250	0.7240	0.5242
-5	3.6	3.9102	3.6250	0.2852	0.0813
0	3.4	3.4714	3.6250	-0.1536	0.0236
5	3.1	3.0326	3.6250	-0.5924	0.3509
10	2.7	2.5938	3.6250	-1.0312	1.0633
19	1.8	1.8040	3.6250	-1.8210	3.3160
				Sum	8.7452

From the data values,

X	Y
-18	5.2	3.6250	1.5750	2.4806
-15	4.7	3.6250	1.0750	1.1556
-10	4.5	3.6250	0.8750	0.7656
-5	3.6	3.6250	-0.0250	0.0006
0	3.4	3.6250	-0.2250	0.0506
5	3.1	3.6250	-0.5250	0.2756
10	2.7	3.6250	-0.9250	0.8556
19	1.8	3.6250	-1.8250	3.3306
				8.9150

The means squares are computed using the formula,

The F-statistic values is obtained using the formula,

The F-critical value

The F-critical value is obtained from F-distribution table for significance level = 0.05, numerator degree of freedom = 1 and denominator degree of freedom = 6

Conclusion,

Since,

The null hypothesis is rejected, hence we can conclude that there is a significant relation between dependent and independent variable.

c)

The significance of slope is tested by calculating t statistic using the formula,

Where,

The P-value for t statistic is obtained from t distribution table for degree of freedom = n - 2 = 6

Since P-value = 0.0000 < 0.05 at 5% significant level, the null hypothesis is rejected

d)

The standard error of slope is,

The 95% confidence interval for slope is obtained using the formula,

the t critical value is obtained using the t distribution table for significance level = 0.05 and degree of freedom = n - 2 = 8 - 2 = 6

e)

The R-square (coefficient of determination) value is obtained using the formula,