When Carson runs the 400 meter dash, his finishing times are normally distributed with a mean of 76 seconds and a standard deviation of 2 seconds. If Carson were to run 16 practice trials of the 400 meter dash, how many of those trials would be between 73 and 75 seconds, to the nearest whole number?
First, we need to calculate the z-scores for 73 seconds and 75 seconds:
For 73 seconds:
z = (73 - 76) / 2 = -1.5
For 75 seconds:
z = (75 - 76) / 2 = -0.5
Next, we need to find the probability of Carson finishing the race between 73 and 75 seconds. This can be done by finding the area under the normal distribution curve between the z-scores of -1.5 and -0.5.
Using a standard normal distribution table or a calculator, we find:
P(-1.5 < Z < -0.5) = 0.6915 - 0.3085 = 0.383
Therefore, approximately 38.3% of Carson's 16 practice trials would have finishing times between 73 and 75 seconds. To find how many trials this represents, we multiply 16 by 0.383:
16 * 0.383 = 6.128
Rounded to the nearest whole number, Carson would have approximately 6 trials between 73 and 75 seconds.