Approximate a Binomial Upper Tail with a Normal Distribution

Let (X\sim\operatorname{Binomial}(100,0.5)). Estimate (P(X>60)) with a normal approximation, including a continuity correction. Also state the exact expression that your approximation is estimating and assess whether the approximation is reasonable. ### Constraints & Assumptions - Interpret (X>60) as (X\ge61). - Use the binomial mean and variance rather than fitting parameters from data. - Report the standardization step and an approximate numerical probability. ### Clarifying Questions to Ask - Is a continuity correction expected? - Is an exact binomial sum required as a benchmark, or only the normal approximation? - What level of numerical precision should be reported? ```hint Move the boundary by one half The continuous normal event corresponding to (X\ge61) begins at (60.5). ``` ### What a Strong Answer Covers - The correct binomial mean (np) and standard deviation (sqrt{np(1-p)}). - A (z)-score based on 60.5, not 60 or 61 without explanation. - The upper-tail probability and the exact finite sum for comparison. - A check of the usual normal-approximation conditions. ### Follow-up Questions - How inaccurate is the answer if the continuity correction is omitted? - What changes if the coin has head probability (0.1)? - When would a saddlepoint or exact computation be preferable?