To determine whether the normal approximation to the binomial distribution can be used for \(X \sim \text{Binomial}(n=50, p=0.001)\), we can use the rule of thumb that the normal approximation is appropriate if both \(np\) and \(n(1-p)\) are greater than or equal to 5.
First, we calculate:
\[ np = n \cdot p = 50 \cdot 0.001 = 0.05 \]
Next, we calculate:
\[ n(1-p) = n \cdot (1-p) = 50 \cdot (1-0.001) = 50 \cdot 0.999 = 49.95 \]
Now we check the conditions:
- \(np = 0.05\) (which is less than 5)
- \(n(1-p) = 49.95\) (which is greater than 5)
Since \(np\) is significantly less than 5, the normal approximation is not appropriate for this situation. The distribution of \(X\) is highly skewed, mainly concentrated around lower values, and using the normal approximation could lead to inaccurate results for calculating \(P(X > 12)\).
In summary, do not use the normal approximation to find \(P(X > 12)\) for this binomial distribution, as \(np < 5\). Instead, consider using exact binomial probabilities or other appropriate methods.
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