Question

NEED HELP ASAP FOR 70PTS

A certain standardized​ test's math scores have a​ bell-shaped distribution with a mean of 515 and a standard deviation of 110.

Complete parts​ (a) through​ (c). ​

(a) What percentage of standardized test scores is between 185 and 845​? nothing​% ​(Round to one decimal place as​ needed.)

​(b) What percentage of standardized test scores is less than 185 or greater than 845​? nothing​% ​(Round to one decimal place as​ needed.)

​(c) What percentage of standardized test scores is greater than 735​? nothing​% ​(Round to one decimal place as​ needed.)

Answer 1

Solution: We are given:

`\mu=515, \sigma=110`

(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:

`z=(x-\mu)/(\sigma)`

`=(185-515)/(110) =-3`

When x=845, we have:

`z=(x-\mu)/(\sigma)`

`=(845-515)/(110) =3`

Therefore, we have to find
`P(-3\leq z \leq 3)`.

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:

`z=(x-\mu)/(\sigma)`

`=(185-515)/(110) =-3`

When x=845, we have:

`z=(x-\mu)/(\sigma)`

`=(845-515)/(110) =3`

Therefore, we have to find
`P(z<-3) +P(z>3)`.

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
`P(z<-3) +P(z>3)` = 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:

`z=(x-\mu)/(\sigma)`

`=(735-515)/(110) =2`

Therefore, we have to find
`P(z>2)`.

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.