Question

the graph shows the distribution of the lengths (in seconds) of videos on a popular video - streaming site. the distribution is approximately normal, with a mean of 264 seconds and a standard deviation of 75 seconds. what percentage of videos on the streaming site are between 264 and 489 seconds? 0.15% 49.85% 95% 99.7%

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean $\mu = 264$ seconds and the standard deviation $\sigma=75$ seconds.
For the lower - bound $x_1 = 264$: $z_1=\frac{x_1-\mu}{\sigma}=\frac{264 - 264}{75}=0$
For the upper - bound $x_2 = 489$: $z_2=\frac{x_2-\mu}{\sigma}=\frac{489 - 264}{75}=\frac{225}{75}=3$

Step2: Use the empirical rule for the normal distribution

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:
The proportion of data within $z = 0$ and $z = 3$ can be found as follows.
The proportion of data within $z=-3$ and $z = 3$ is approximately $99.7\%$, and the proportion of data within $z=-3$ and $z = 0$ is the same as the proportion of data within $z = 0$ and $z = 3$.
Since the normal distribution is symmetric about the mean ($z = 0$), the proportion of data within $z = 0$ and $z = 3$ is $\frac{99.7\%}{2}=49.85\%$

Answer:

49.85%