A chemist is calibrating a spectrophotometer that will be used to measure the concentration of carbon monoxide (CO) in atmospheric samples. To check the calibration, samples of known concentration are measured. The true concentrations (x) and the measured concentrations (y) are given in the following table. Because of random error, repeated measurements on the same sample will vary. The machine is considered to be in calibration if its mean response is equal to the true concentration.
True concentration
(ppm)
|
Measured concentration
(ppm)
|
0
|
1
|
10
|
11
|
20
|
21
|
30
|
28
|
40
|
37
|
50
|
48
|
60
|
56
|
70
|
68
|
80
|
75
|
90
|
86
|
100
|
96
|
To check the calibration, the linear model y = β0 + β1x + ε is fit. Ideally, the value of β0 should be 0 and the value of β1 should be 1.
a. Compute the least-squares estimates
and
.
b. Can you reject the null hypothesis H0 : β0 = 0?
c. Can you reject the null hypothesis H0 : β1 = 1?
d. Do the data provide sufficient evidence to conclude that the machine is out of calibration?
e. Compute a 95% confidence interval for the mean measurement when the true concentration is 20 ppm.
f. Compute a 95% confidence interval for the mean measurement when the true concentration is 80 ppm.
g. Someone claims that the machine is in calibration for concentrations near 20 ppm. Do these data provide sufficient evidence for you to conclude that this claim is false? Explain.