Question

the graph shows the distribution of the length (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 114 and 489 seconds? 50% 68% 97.35% 99.7%

Explanation:

Step1: Calculate the number of standard deviations from the mean

For \(x = 114\): \(z_1=\frac{114 - 264}{75}=\frac{- 150}{75}=-2\)
For \(x = 489\): \(z_2=\frac{489 - 264}{75}=\frac{225}{75}=3\)

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

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data is within \(z=\pm1\)
• Approximately \(95\%\) of the data is within \(z = \pm2\)
• Approximately \(99.7\%\) of the data is within \(z=\pm3\)

The area to the left of \(z=- 2\) is \(0.0228\) (from the standard normal table) and the area to the left of \(z = 3\) is \(0.9987\)
The percentage of data between \(z=-2\) and \(z = 3\) is \(P(-2<Z<3)=P(Z < 3)-P(Z<-2)=0.9987 - 0.0228=0.9759\approx97.35\%\)

Answer:

97.35%