Average fetal heartrate is $140$ bpm, with standard deviation of $12$ bpm. What is the probability that a randomly chosen fetal heartrate differs from the mean by more than $25$ bpm?
Can this problem be solved using central limit theorem?
The theorem gives us information about the means of samples not about the probability density function, if the question were "what is the probability that the average fetal heart rate of a population of $40$ people differs from the mean by more than $25$ bpm" it would have been easy to solve, but I don't really know how to approach this problem!
Is my understanding correct? and how would you solve this problem?
We can just calculate the $Z$-score here. As our $x$ value here pertains to when the fetal heartrate differs from the mean by $\gt25$ bpm, this can be interpreted as $x=140+25=165$. Hence
\begin{align*} Z&=\frac{x-\mu}{\sigma}\\ &=\frac{165-140}{12}\\ &=2.083\\ \end{align*}
Therefore
\begin{align*} \mathbb{P}(X\gt165)&=\mathbb{P}(Z\gt2.083)\\ &=1-\mathbb{P}(Z\lt2.083)\\ &=1-\Phi(2.083)\\ &=1−0.98124\\ &\approx0.01876\\ \end{align*}
Note that you don't have enough information to calculate the Central limit theorem. We would need a sample size $n$ for a start. More information can be found here.