For this problem, we are given a normal distribution that has a mean equal to $1800 and a standard deviation of $135. We need to calculate a few percentages based on this information.
The first one is the approximate percentage of buyers who paid more than 2070.
In order to solve this, we need to find the z-score, which is given by the following formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]Where "x" is the desired value, "mi" is the mean, and "sigma" is the standard deviation. For this case we have:
[tex]z=\frac{2070-1800}{135}=2[/tex]The z-score is equal to 2. Now we need to go on the z-table where we will find the percentage of everyone that paid less than 2070, this is not the value we want, but we can subtract this value from 1 to determine what we want. This is shown below:
[tex]P(z>2)=1-P(z<2)=1-0.9772=0.02275[/tex]The percentage is 2.275%
Now we need to find the percentage of buyers who paid between 1800 and 2205. For this we need to find the z-score for both values:
[tex]\begin{gathered} z_1=\frac{1800-1800}{135}=0 \\ z_2=\frac{2205-1800}{135}=3 \end{gathered}[/tex]To solve this item, we need to go on the z-table and find the percentage for both. Then we need to subtract the P(z<3) with P(z<0).
[tex]P(0The percentage is 49.87%Now we need to calculate the values between 1800 and 2070. We already found the score for 1800 and the one for 2070, so we can subtract the percentages directly:
[tex]P(0The percentage is 47.72%Now we need to calculate the percentage that paid less than $1395. We need to find the z-score for this value:
[tex]z=\frac{1395-1800}{135}=-3[/tex]Now we need to find the value that corresponds to -3 on the z-table:
[tex]P(z<-3)=0.0013[/tex]The percentage is 0.13%
Now we need to find the percentage of buyers who paid between $1665 and $1800. We already have the percentage for those who paid less than $1800, now we need to calculate the z-score of $1665.
[tex]z=\frac{1665-1800}{135}=-1[/tex]The percentage is:
[tex]P(-1The percentage is 34.13%
