Solution: We are given:
[tex]\mu=515, \sigma=110[/tex]
(a) What percentage of standardized test scores is between 185 and 845?
In order to find the percentage of scores that fall between 185 and 845, we use the z score formula first:
When x = 185, we have:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{185-515}{110} =-3[/tex]
When x=845, we have:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{845-515}{110} =3[/tex]
Therefore, we have to find [tex]P(-3\leq z \leq 3)[/tex].
From the empirical rule of normal distribution 99.7% of data falls within 3 standard deviation's from mean.
Therefore, 99.7% of standardized test scores is between 185 and 845.
(b) What percentage of standardized test scores is less than 185 or greater than 845?
In order to find the percentage of scores that is less than 185 or greater than 845, we use the z score formula first:
When x = 185, we have:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{185-515}{110} =-3[/tex]
When x=845, we have:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{845-515}{110} =3[/tex]
Therefore, we have to find [tex]P(z<-3) +P(z>3)[/tex].
From the empirical rule of normal distribution 0.15% of data falls 3 standard deviation's below mean and 0.15% of data falls 3 standard deviation's above mean.
Therefore [tex]P(z<-3) +P(z>3)[/tex] = 0.15% +0.15% =0.3%
Therefore, 0.3% of standardized test scores is less than 185 or greater than 845.
(c) What percentage of standardized test scores is greater than 735?
Answer: In order to find the percentage of scores that is greater than 735, we use the z score formula first:
When x = 735, we have:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{735-515}{110} =2[/tex]
Therefore, we have to find [tex]P(z>2)[/tex].
From the empirical rule of normal distribution 2.5% of data falls 2 standard deviation's above mean.
Therefore, 2.5% of standardized test scores is greater than 735.
