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The finishing times for a running competition follow a normal distribution. Given,Mean, µ = 490 secondsStandard Deviation, σ = 50 secondsWe need to find the time needed to finish in the fastest 10% of runners in this competition.
The fastest 10% of runners is nothing but the 90th percentile of the finishing times.For a normally distributed variable, we can find the z-score of the percentile and convert it back to the value of the variable using the mean and standard deviation of the distribution.The formula for the z-score is:z = (x - µ) / σFor the 90th percentile, the corresponding z-score can be found using the standard normal distribution table or calculator. Using the standard normal table, we get:z = 1.28 (approx)Using the formula, we can find the value of x as:x = zσ + µx = 1.28 × 50 + 490x = 646 secondsThis is the finishing time for the fastest 10% of runners in the competition.To find the time needed to finish in the fastest 10% of runners in this competition, we first need to identify the 90th percentile of the finishing times. For normally distributed variables, we can find the z-score of the percentile and convert it back to the value of the variable using the mean and standard deviation of the distribution.In this case, the mean is 490 seconds and the standard deviation is 50 seconds. The z-score corresponding to the 90th percentile can be found using the standard normal distribution table or calculator.Using the standard normal table, we find that the z-score for the 90th percentile is 1.28 (approx). Using the formula for the z-score, we can find the value of x as:x = zσ + µx = 1.28 × 50 + 490x = 646 secondsTherefore, the finishing time needed to be in the fastest 10% of runners in this competition is 646 seconds. The options given in the question are: a. 451 sec b. 432 sec c. 426 sec d. 443 sec e. 554 sec. None of these options match our answer. Hence, the correct answer is not given in the options provided.The finishing times for a running competition follow a normal distribution. To find the time needed to finish in the fastest 10% of runners, we need to find the value of the 90th percentile. For normally distributed variables, we can find the z-score corresponding to the percentile using the standard normal distribution table or calculator. We can then convert the z-score back to the value of the variable using the mean and standard deviation of the distribution. In this case, the finishing time needed to be in the fastest 10% of runners is 646 seconds.
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