In: Statistics and Probability
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
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
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