Puzzle Completion Probability Analysis
The probability that Jack will finish before Sam is approximately 0.0004. The probability that Jack's average time on 4 puzzles is less than Sam's average time is approximately 0.1151.
Probability of Jack Finishing Before Sam
To find the probability that Jack will finish before Sam, we need to compare their individual completion times. Jack's completion times are normally distributed with a mean of 500 minutes and a standard deviation of 35 minutes. Sam's completion times are normally distributed with a mean of 475 minutes and a standard deviation of 25 minutes.
Using the normal distribution, we can calculate the probability that Jack will finish before Sam by finding the area under the curve to the left of the value 112 (Jack's completion time). Using a normal distribution table or calculator, we find that the probability is approximately 0.0004.
Probability of Jack's Average Time on 4 Puzzles Being Less Than Sam's
To find the probability that Jack's average time on 4 puzzles is less than Sam's average time, we need to consider the distribution of sample means. The mean of the sample means is equal to the population mean, and the standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size.
Since we are comparing the average time on 4 puzzles, the sample size is 4. Jack's average time on 4 puzzles is normally distributed with a mean of 500 minutes and a standard deviation of 35 minutes divided by the square root of 4 (which is 2). Sam's average time on puzzles is normally distributed with a mean of 475 minutes and a standard deviation of 25 minutes divided by the square root of 4 (which is 2).
Using the normal distribution, we can calculate the probability that Jack's average time on 4 puzzles is less than Sam's average time by finding the area under the curve to the left of the value of Sam's average time. Using a normal distribution table or calculator, we find that the probability is approximately 0.1151.