When Natalie runs the 400 meter dash, her finishing times are normally distributed with a mean of 71 seconds and a standard deviation of 2.5 seconds. What percentage of races will her finishing time be between 71 and 77 seconds, to the nearest tenth?
Answer
3. Calculate the percentage of races between 71 and 77 seconds: \((P(Z_2) - P(Z_1))\times 100 = (0.9918 - 0.5)\times 100 = 49.2\%\)
Step 1 ：1. Determine Z-scores: \(Z_1 = \frac{71 - 71}{2.5} = 0\), and \(Z_2 = \frac{77 - 71}{2.5} = 2.4\)
Step 2 ：2. Look up Z-scores in the standard normal table or use a calculator to find the cumulative probabilities: \(P(Z_1) = 0.5\), \(P(Z_2) = 0.9918\)
Step 3 ：3. Calculate the percentage of races between 71 and 77 seconds: \((P(Z_2) - P(Z_1))\times 100 = (0.9918 - 0.5)\times 100 = 49.2\%\)
