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 for normal distributions

The empirical rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\))
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\))

The proportion of data between \(z=-2\) and \(z = 3\) can be found as follows:
The proportion of data less than \(z=-2\) is \(0.0228\) (from the standard normal table)
The proportion of data less than \(z = 3\) is \(0.9987\) (from the standard normal table)
The proportion of data between \(z=-2\) and \(z = 3\) is \(0.9987-0.0228 = 0.9759\approx97.35\%\)

Answer:

\(97.35\%\)