Question

Many studies have been conducted to determine the concentration of CO (carbon monoxide) near freeways with various conditions of traffic flow. The basic technique involves measuring CO concentrattions with a spectrophotometer. These machines are quite delicate and have to be calibrated every day. If the machine reads close to 70 ppm on the span gas, it’s ready for use; if not, it has to be adjusted.
Each data set below represents some readings on span gas. Assume the Gaussian model, with errors following the normal curve. In each case, make a t-test to see whether the instrument is properly calibrated or not.
a) 71, 68, 79
b) 71, 68, 79, 84, 78, 85, 69
c) 71
d) 71, 84

Answer 1

To know, if the calibration is proper or not, we will run a 1 sample-t test

Null hypothesis: There is no significant difference between the mean reading of the machine and 70 ppm

Alternate hypothesis: There is a significant difference between the mean reading of the machine and 70 ppm

	Readings
	71
	68
	79
	71
	68
	79
	84
	78
	85
	69
	71
	71
	84
sample size (n)	13
Mean (µ)	75.2
Standard deviation (σ)(s)	6.5
df (n-1)	12

Assuming confidence level of 95%

with CI of 95% and df of 12, t value as per below table is 1.7832

Table of t-Distribution Areas df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.82 63.657127.321 318.309 636.619 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14089 22.327 31.599 0.215 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3 0.765 0.978 1.250 1.638 2.3533.182 3.482 4.541 5.841 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 7.453 10.215 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.222 3.646 5.598 4.773 4.317 4.029 3.833 3.690 3.581 5 0.727 0.920 1.156 1.476 2.015 2.57 2.757 3.365 4.032 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.7643.169 11 0.697 0.876 1.088 1.363 1.796 2.2012.328 2.718 3.106 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 13 0.694 0.870 1.079 1.350 77 2.160 2.282 2.650 3.012 3.497 3.428 3.372 14 0.692 0.868 1.076 1.345 1.76 2.145 2.264 2.624 2.977 3.326 3.286 3.252 15 0.691 0.866 1.074 1.34 1.753 2.13 2.249 2.602 2.947 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 18

We get below values

One-Sample T: Readings

Descriptive Statistics

N	Mean	StDev	SE Mean	95% CI for μ
13	75.23	6.46	1.79	(71.33, 79.13)

μ: mean of Readings

Test

Null hypothesis	H₀: μ = 70
Alternative hypothesis	H₁: μ ≠ 70

T-Value	P-Value
2.92	0.013

Since p-value is less than the level of significance, we reject the null hypothesis and conclude that there is a significant difference between the mean reading of the machine and 70 ppm

Hence the machine needs calibration