The empirical rule states that:
→ 68% of any normal distribution is within one standard deviation from its mean, you can express it as follows:
[tex]\mu\pm\sigma=68\%[/tex]
→ 95% of the normal distribution is found within 2 standard deviations of the mean, you can express this as:
[tex]\mu\pm2\sigma=95\%[/tex]
→ 99.5% of the normal distribution is within 3 standard deviations of the distribution, you can express this as:
[tex]\mu\pm3\sigma=99.5\%[/tex]
Remember that the mean is the center of the normal distribution, which means that 50% of the normal distribution is found below the mean and the other 50% of the distribution is above the mean.
Considering the given normal distribution:
The value at the center of the distribution is 20, which means that the mean time of this normal distribution is 20 minutes.
Assuming that the standard deviation of the distribution is σ= 5 minutes, then:
12. The times 15 minutes and 25 minutes are within 1 standard deviation:
If μ=20 min and σ=5 min, then
[tex]\begin{gathered} μ\pmσ=68\% \\ 20\pm5=68\% \end{gathered}[/tex]
Then 68% of the students will wait between 15 and 25 minutes.
13. You have to determine the percentage of students that will wait less than 20 minutes. As mentioned before, the mean of this distribution is μ=20 min and the mean is the exact center of it, which means that it divides it into two halves.
With this in mind, you can conclude that 50% of the students will wait less than 20 minutes.
14. You have to determine the percentage of students that will wait more than 35 minutes.
Following the parameters μ=20 min and σ=5, the value of 35 minutes can be considered to be 3 standard deviations from the mean.
Following the empirical rule, we know that 99.5% of the distribution is within 3 standard deviations from the mean, this means that you will find 99.5% of the distribution between 5 and 35 minutes, the remaining 0.5% of the distribution will be equally divided below and above these values:
If you look at the graph, only 0.25% of the distribution is above the time of 35 min, which means that 0.25% of the students will wait more than 35 minutes.
15. There are 5,300 students coming to the center. You have to determine how many of them will wait more than 35 minutes, to do so, you have to use the percentage determined on the previous item.
It was already established that 0.25% of the students will wait more than 35 minutes, the number of students that will wait more than 35 minutes is equal to 0.25% of the total number of students that visit the advising center.
Calculate the 0.25% of 5,300:
[tex]\frac{(5,300*0.25)}{100}=13.25[/tex]
13 students will wait for more than 35 minutes.
