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Sensitivity and Specificity Definitions:
| Measure | Formula | Meaning |
|---|---|---|
| Sensitivity | \(\frac{TP}{TP + FN}\) | % of true positives correctly identified (rules OUT disease if negative โ SnNOut) |
| Specificity | \(\frac{TN}{TN + FP}\) | % of true negatives correctly identified (rules IN disease if positive โ SpPIn) |
- 95% sensitivity: The tool correctly identifies 95% of patients who have the condition
- 60% specificity: Only 60% of those without the condition are correctly identified as negative โ meaning a high false positive rate (40%)
1. Carriage Out: Debit side too high by $90 (Correction: Credit Suspense $90).
2. Purchases Returns: Credit side (returns) too high by $100 (Correction: Debit Suspense $100).
3. Wrong Person: No effect on Suspense (Debit/Credit match).
4. Rent: Income ($350 Cr) put in Expense ($350 Dr). Total error is $700 on debit side (Correction: Credit Suspense $700).
Net correction: $90 (Cr) - $100 (Dr) + $700 (Cr) = $690 Credit. To clear this, the opening balance must have been $690 Credit.
Find all values of $\lambda$ for which the system $2x_1-2x_2+x_3=\lambda x_1$, $\ 2x_1-3x_2+2x_3=\lambda x_2$, $\ {-x_1+2x_2=\lambda x_3}$ has a non-trivial solution.
Rewrite as $(A-\lambda I)X=0$. For non-trivial solutions, $\det(A-\lambda I)=0$.
$A = \begin{pmatrix}2&-2&1\\2&-3&2\\-1&2&0\end{pmatrix}$
The characteristic equation simplifies to $\lambda^3 + \lambda^2 - 5\lambda + 3 = 0$, which factors as $(\lambda-1)^2(\lambda+3)=0$.
Roots: $\lambda=1$ (repeated) and $\lambda=-3$. So the set contains exactly two elements: $\{1, -3\}$.
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