2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.
| Patient | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
| 160 | 390 | 169 | 175 | 125 | 420 | 171 | 250 | 210 | 258 | 186 | 243 |
$$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$
- Find the mean and standard deviation of the level of cotinine in a patient's blood.
- Find the median, upper and lower quartiles of these data.
A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
- Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly.
Research suggests that cotinine levels in the blood form a skewed distribution.
One measure of skewness is found using \(\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }\). - Evaluate this measure and describe the skewness of these data.