Question

normal distribution - empirical rule
question
the amount of time a certain brand of light bulb lasts is normally distributed with a mean of 1500 hours and a standard deviation of 65 hours. using the empirical rule, what percentage of light bulbs last between 1305 hours and 1695 hours?
answer
attempt 1 out of 2

Explanation:

Step1: Find how many standard deviations from the mean.

First, calculate the number of standard deviations for 1305 and 1695 from the mean (1500). For 1305: $1500 - 1305 = 195$, then $\frac{195}{65}=3$. For 1695: $1695 - 1500 = 195$, then $\frac{195}{65}=3$. So both values are 3 standard deviations from the mean.

Step2: Apply the empirical rule.

The empirical rule states that for a normal distribution, about 99.7% of the data lies within 3 standard deviations of the mean (i.e., between $\mu - 3\sigma$ and $\mu + 3\sigma$). Here, $\mu = 1500$, $\sigma = 65$, so 1305 is $\mu - 3\sigma$ and 1695 is $\mu + 3\sigma$. Thus, the percentage of light bulbs lasting between 1305 and 1695 hours is 99.7%.

Answer:

99.7%