1. **Problem statement:** Bilbo wants to test at the 5% significance level which number of transactions starting with digit 1 would be suspicious, given that the expected proportion is 0.30 (Benford's Law) in 100 transactions. 2. **Formula and approach:** We model the number of transactions starting with 1 as a binomial random variable $X \sim \text{Binomial}(n=100, p=0.3)$. 3. **Normal approximation:** Since $n$ is large, approximate $X$ by a normal distribution: $$ X \approx N(\mu, \sigma^2) \quad \text{where} \quad \mu = np = 100 \times 0.3 = 30, \quad \sigma = \sqrt{np(1-p)} = \sqrt{100 \times 0.3 \times 0.7} = \sqrt{21} \approx 4.583. $$ 4. **Significance level and critical values:** At 5% significance level (two-tailed), the critical z-values are approximately $\pm 1.96$. 5. **Calculate critical values for $X$:** $$ X_{lower} = \mu - 1.96 \times \sigma = 30 - 1.96 \times 4.583 \approx 30 - 8.98 = 21.02 $$ $$ X_{upper} = \mu + 1.96 \times \sigma = 30 + 8.98 = 38.98 $$ 6. **Apply continuity correction:** Lower bound: $21.5$ (since $X$ is discrete, use $21.5$ to include 22 and above) Upper bound: $38.5$ 7. **Interpretation:** - If the number of transactions starting with 1 is less than or equal to 21, or greater than or equal to 39, Bilbo should be suspicious. **Final answer:** Bilbo should be suspicious if the number of transactions starting with 1 is $\leq 21$ or $\geq 39$ at the 5% significance level.