Question

1. when ruby runs the 400 meter dash, her finishing times are normally distributed with a mean of 89 seconds and a standard deviation of 3 seconds. using the empirical rule what percentage of races will her finishing time be between 86 and 92 seconds? draw and label a bell curve.

Explanation:

Step1: Identify the mean and standard deviation

The mean ($\mu$) is 89 seconds, and the standard deviation ($\sigma$) is 1 second (assuming the "a 1 second" is a typo and should be "1 second"). We need to find how many standard deviations 86 and 92 are from the mean.

For 86: $z_1=\frac{86 - 89}{1}=- 3$

For 92: $z_2=\frac{92 - 89}{1}=3$

Step2: Apply the empirical rule

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately 68% of data lies within $\mu\pm\sigma$
• Approximately 95% of data lies within $\mu\pm2\sigma$
• Approximately 99.7% of data lies within $\mu\pm3\sigma$

Since our values are at $\mu - 3\sigma$ (86) and $\mu+3\sigma$ (92), the percentage of data between them is approximately 99.7%.

(For the bell curve: Draw a normal distribution curve. Label the mean (89) at the center. Mark $\mu-\sigma = 88$, $\mu + \sigma=90$, $\mu-2\sigma = 87$, $\mu + 2\sigma=91$, $\mu-3\sigma = 86$, $\mu + 3\sigma=92$ on the horizontal axis. Shade the area between 86 and 92, which corresponds to about 99.7% of the area under the curve.)

Answer:

The percentage of races with finishing times between 86 and 92 seconds is approximately 99.7%.