The question goes as follows:
An airline reports that $77.5\%$ of its flights arrive on time. A check of $200$ randomly selected airline flights shows that $140$ of them arrived on time. Estimate the probability that among $200$ airline flights, $140$ or fewer arrive on time. Based on this result, does it seem plausible that the claimed on-time rate of $77.5\%$ could be correct?
I have tried to workout the problem and ended up with a probability of $0.0055$ ($Z$ score is $-2.54$) Given I worked out these values: $n=200$, $p= 0.775$, and $q= 0.225$, the mean is $155$, and the standard deviation is: $5.906$.
My question(s) are:
Am I going about this question correctly assuming that $n>30$ (central limit theorem)?
And how does one know when to use the greater than/greater than or equal...less than/less than or equal to signs when working these problems?
I had used $P(x \leq 140)$ while setting up this particular problems, but I am still somewhat struggling on how to correctly interpret word problems. Thank you for your time.
The approximate probability of $140$ or less flights arriving on time, assuming a Normal($155, 5.906^2)$ distribution is: $$\mathsf P(x < 140.5)\approx 0.339$$
Note: the $.5$ adend is because our variable actually realises discrete (integer) values while the Normal approximation is a continuous distribution; so we cover the pre-rounding values.