91Ó°ÊÓ

To find the probability that the reaction time exceeds 0.3 second, the z-score for this value must be calculated first. The z-score for a given measurement is given by the formula: \(Z = \frac{(X - \mu)}{\sigma}\) where \(X\) is the value of interest, \(\mu\) is the mean and \(\sigma\) is the standard deviation. In this case, \(X=0.3\)s, \(\mu=0.5\)s and \(\sigma=0.4\)s. Plugging in these values yields: \(Z = \frac{(0.3 - 0.5)}{ 0.4} = -0.5\).

The z-score allows to calculate the probability for a given value of x by referring to a standard normal table or using a calculator with a built-in function. The z-score of -0.5 corresponds to a cumulative probability of 0.3085, which represents the proportion of times the reaction time would be less than 0.3 second. To find the probability that the reaction time is more than 0.3 seconds, subtract that value from 1: \(P(X > 0.3) = 1 - P(X \leq 0.3) = 1 - 0.3085 = 0.6915.\) So, the probability that a reaction time from the pilot takes more than 0.3 seconds is approximately 0.6915 or 69.15%.

The second part of the problem asks which reaction time is exceeded 95% of the time. This can be reinterpreted as finding the 5th percentile (since percentiles are always referred from below, 5% from below is 95% from above). In order to find this, it is necessary to derive the z-score from the standard normal table corresponding to the 0.05 cumulative probability. This z-score is approximately -1.645. The corresponding x value can be found using the formula: \(X = \mu + Z\sigma\), where Z is the z-score, \(\mu\) is the mean and \(\sigma\) is the standard deviation. Plugging in the appropriate values gives: \(X = 0.5 + (-1.645*0.4) = 0.5 - 0.658 = -0.158\) s. However, time can't be negative. Seems like the standard deviation is quite large compared to the mean, which causes the issue. That means practically it's impossible to have 95% of the time to have reaction time exceeded.

Make sure to take a step back and ask if these numbers make sense in the context of the initial problem. The first probability suggests that for a randomly selected trial, the pilot would take more than 0.3 seconds about 69.15% of the time, which seems quite feasible in the context of reaction times. The time which is exceeded 95% of the time is negative because the distribution is vast. We can interpret this as practically all the time the reaction will occur since we can't have a negative time.