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 39 and 189 seconds? 0.15% 15.85% 16% 34%

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean $\mu = 264$ and standard deviation $\sigma=75$.
For $x = 39$: $z_1=\frac{39 - 264}{75}=\frac{- 225}{75}=-3$
For $x = 189$: $z_2=\frac{189 - 264}{75}=\frac{- 75}{75}=-1$

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that for a normal distribution:

• Approximately $68\%$ of the data lies within $\mu\pm\sigma$ ($z=- 1$ to $z = 1$)
• Approximately $95\%$ of the data lies within $\mu\pm2\sigma$ ($z=-2$ to $z = 2$)
• Approximately $99.7\%$ of the data lies within $\mu\pm3\sigma$ ($z=-3$ to $z = 3$)

The percentage of data from $z=-3$ to $z=-1$:
The percentage from $z = -3$ to $z = 3$ is $99.7\%$, from $z=-2$ to $z = 2$ is $95\%$, from $z=-1$ to $z = 1$ is $68\%$.
The percentage from $z=-3$ to $z=-2$ is $\frac{99.7\% - 95\%}{2}=2.35\%$
The percentage from $z=-2$ to $z=-1$ is $\frac{95\% - 68\%}{2}=13.5\%$
The percentage from $z=-3$ to $z=-1$ is $2.35\%+13.5\% = 15.85\%$

Answer:

15.85%