Question

when landon runs the 400 meter dash, his finishing times are normally distributed with a mean of 89 seconds and a standard deviation of 1 second. using the empirical rule, determine the interval of times that represents the middle 95% of his finishing times in the 400 meter race. answer attempt 1 out of 2

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule (also known as the 68 - 95 - 99.7 rule) states that for a normal distribution, approximately 95% of the data lies within \( \mu\pm2\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

Step2: Identify the values of \( \mu \) and \( \sigma \)

Given that the mean \( \mu = 89 \) seconds and the standard deviation \( \sigma=1 \) second.

Step3: Calculate the lower and upper bounds of the interval

The lower bound is \( \mu - 2\sigma \) and the upper bound is \( \mu+ 2\sigma \).

Substitute \( \mu = 89 \) and \( \sigma = 1 \) into the formulas:

Lower bound: \( 89-2\times1=89 - 2=87 \)

Upper bound: \( 89 + 2\times1=89+ 2 = 91 \)

Answer:

The interval of times that represents the middle 95% of his finishing times is from 87 seconds to 91 seconds, or in interval notation \( (87, 91) \) (or \( [87, 91] \) since the distribution is continuous in this context).