Question

Suppose we have two thermometers. One thermometer is very precise but is delicate and heavy (X). We have another thermometer that is much cheaper and lighter, but of unknown precision (Y). We would like to know if we can (reliably) bring the lighter thermometer with us into the field. So, we set up an experiment where we expose both thermometers to 31 different temperatures and measure the temperature with each. We get the following observations

x = 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120
y = 0.02, 3.99, 7.91, 12.03, 16.09, 20.00, 23.98, 28.09, 31.94, 36.03, 40.00, 44.05, 47.95, 52.00, 55.87, 59.90, 63.91, 67.95, 72.11, 76.02, 80.01, 84.10, 88.06, 91.74, 96.02, 99.95, 103.87, 108.01, 111.99, 116.04, 120.03

We want to decide if these thermometers seem to be measuring the same temperatures. Let's use the thershold α= 0.1.

Required:
Write down the appropriate hypothesis tests for β1.

Answer 1

a)H0: β1= 1 and Ha: β1 ≠1

b) The test statistic is____.t= b- β/ sb

c) The p-value is____.0.999144

(d) Therefore, we can conclude that:_____. there is not enough evidence to reject the null hypothesis.

The data provides no evidence at the 0.1 significance level that these thermometers are not consistent.

Explanation:

This is called testing hypotheses about β, the population regression co efficient.

The null and alternate hypotheses are

H0: β1= 1 and Ha: β1 ≠1

The significance level is ∝=0.1 and ∝/2 is 0.05

The critical region at t ∝/2 (29)= t ≤ - 1.699 , t ≥ 1.699

The test statistic is

t= b- β/ sb

which has t distribution with υ= 31-2= 29 degrees of freedom.

Calculations

Ŷ = a +bX

b = SPxy /SS x= Σ(xi-x`)(yi-y`)/Σ(xi-x`)²

b = 39671.6/39680= 0.9998

a = y` - bx`

x`= 60

y`= 59.989

a = 59.989 -0.9998*60 = 0.001734

Syx²= Σ( yi -y`)²/ n-2= 39663.3857/ 29=1367.68965

Syx= 36.9822

Sb= Syx/ √∑ (x-x`)²

Sb= 36.9823/ √39680

Sb= 36.9823/ 199.984

Sb= 0.18493

x-x` y-y` (x-x`)² (x-x`)(y-y`)

-60 -59.969 3600 3598.1419

-56 -55.999 3136 3135.9458

-52 -52.079 2704 2708.1097

-48 -47.959 2304 2302.0335

-44 -43.899 1936 1931.5574

-40 -39.989 1600 1599.5613

-36 -36.009 1296 1296.3252

-32 -31.899 1024 1020.769

-28 -28.049 784 785.3729

-24 -23.959 576 575.0168

-20 -19.989 400 399.7806

-16 -15.939 256 255.0245

-12 -12.039 144 144.4684

-8 -7.989 64 63.9123

-4 -4.119 16 16.4761

0 -0.08903 0 0

4 3.921 16 15.6839

8 7.961 64 63.6877

12 12.121 144 145.4516

16 16.031 256 256.4955

20 20.021 400 400.4194

24 24.111 576 578.6632

28 28.071 784 785.9871

32 31.751 1024 1016.031

36 36.031 1296 1297.1148

40 39.961 1600 1598.4387

44 43.881 1936 1930.7626

48 48.021 2304 2305.0065

52 52.001 2704 2704.0503

56 56.051 3136 3138.8542

60 60.041 3600 3602.4581

∑0 0 39680 (SSx) 39671.6 (SPxy)

Putting the values

The test statistic is

t= b- β/ sb

t= 0.9998-1/ 0.18493

t=-0.0010815

Since the calculated t=-0.0010815 does not lie in the critical region t ∝/2 (29)= t ≤ - 1.699 , t ≥ 1.699 we conclude that these thermometers seem to be measure temperatures.

The p-value is ≈ 0.999144

there is not enough evidence to reject the null hypothesis.