A study was conducted on the reaction times of long-distance truck drivers after two hours of non-stop driving. It was found that the reaction times were Normally distributed with an average reaction time of 1.4 seconds and a variance of 0.0625.

Question 12:
What is the probability that a particular truck driver who has been driving for two hours non-stop has a reaction time of more than 2 seconds?

Question 13:
If a sample of 20 truck drivers who have been driving for two hours non-stop is randomly selected, what is the probability that their average reaction time is between 1.35 seconds and 1.4 seconds?

The probability that a truck driver's reaction time is more than 2 seconds is 0.82%. The probability that the average reaction time of a sample of 20 drivers is between 1.35 and 1.4 seconds is 31.59%.

Explanation:

For question 12, we first need to calculate the standard deviation, which is the square root of the variance. So, sqrt(0.0625) = 0.25 seconds. The z-score for a reaction time of 2 seconds is calculated as (2-1.4)/0.25 = 2.4. Consulting a standard z-table, we find the probability corresponding to a z-score of 2.4 is approximately 0.9918. However, since we are interested in reaction times more than 2 seconds, we need to subtract this value from 1. So, the probability that a trucker's reaction time is more than 2 seconds is about 1 - 0.9918 = 0.0082 or 0.82%.

For question 13, the standard deviation of a sample mean is given by the standard deviation divided by the square root of the sample size (in this case, sqrt(20)). This gives us approximately 0.0559. The z-scores for reaction times of 1.35 and 1.4 seconds are -0.9 and 0 respectively. The corresponding probabilities are approximately 0.1841 and 0.5 respectively. Therefore, the probability that the average reaction time is between 1.35 and 1.4 seconds is: 0.5 - 0.1841 = 0.3159 or 31.59%.

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