Ronald scores 700 on the math section of the SAT exam. The distribution of SAT scores is approximately normal with a mean of 500 and a standard deviation of 100. Rubin takes the ACT math exam and scores 31 on the math portion. ACT scores are approximately normally distributed with a mean of 18 and a standard deviation of 6.7
(a) Relative to their peers who also took the tests, who performed better on his test? Explain.
(b) A certain school will only consider those students who score in the top 1% in the math section. What grades would Ronald and Rubin have to receive on their respective tests to be considered for admission?
(c) Between what two grades does 95% of the population fall for the ACT and the SAT exams?

Answer:

a) Due to the higher z-score, Ronald performed better relative to his peers on the test.

b) Ronald needed a grade of at least 732.5, and Rubin of at least 33.58.

c) 95% of the population fall between graded of 4.868 and 31.132 on the ACT.

95% of the population fall between graded of 304 and 696 on the SAT.

Step-by-step explanation:

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question:

(a) Relative to their peers who also took the tests, who performed better on his test? Explain.

We have to find whoever has the higher z-score.

Ronald:

Ronald scores 700 on the math section of the SAT exam. The distribution of SAT scores is approximately normal with a mean of 500 and a standard deviation of 100. So the z-score is found when [tex]X = 700, \mu = 500, \sigma = 100[/tex]. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{700 - 500}{100}[/tex]

[tex]Z = 2[/tex]

Rubin:

Rubin takes the ACT math exam and scores 31 on the math portion. ACT scores are approximately normally distributed with a mean of 18 and a standard deviation of 6.7. So the z-score is found when [tex]X = 31, \mu = 18, \sigma = 6.7[/tex]. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{31 - 18}{6.7}[/tex]

[tex]Z = 1.94[/tex]

Due to the higher z-score, Ronald performed better relative to his peers on the test.

(b) A certain school will only consider those students who score in the top 1% in the math section. What grades would Ronald and Rubin have to receive on their respective tests to be considered for admission?

They have to be in the 100 - 1 = 99th percentile, that is, they need a z-score with a pvalue of at least 0.99. So we need to find for them X when Z = 2.325.

Ronald:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]2.325 = \frac{X - 500}{100}[/tex]

[tex]X - 500 = 232.5[/tex]

[tex]X = 732.5[/tex]

Rubin:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]2.325 = \frac{X - 18}{6.7}[/tex]

[tex]X - 18 = 15.58[/tex]

[tex]X = 33.58[/tex]

Ronald needed a grade of at least 732.5, and Rubin of at least 33.58.

(c) Between what two grades does 95% of the population fall for the ACT and the SAT exams?

They fall between the 100 - (95/2) = 2.5th percentile and the 100 + (95/2) = 97.5th percentile, that is, they fall between X when Z = -1.96 and X when Z = 1.96.

ACT:

Lower bound:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.96 = \frac{X - 18}{6.7}[/tex]

[tex]X - 18 = -1.96*6.7[/tex]

[tex]X = 4.868[/tex]

Upper bound:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.96 = \frac{X - 18}{6.7}[/tex]

[tex]X - 18 = 1.96*6.7[/tex]

[tex]X = 31.132[/tex]

95% of the population fall between graded of 4.868 and 31.132 on the ACT.

SAT:

Lower bound:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.96 = \frac{X - 500}{100}[/tex]

[tex]X - 500 = -196[/tex]

[tex]X = 304[/tex]

Upper bound:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.96 = \frac{X - 500}{100}[/tex]

[tex]X - 500 = 196[/tex]

[tex]X = 696[/tex]

95% of the population fall between graded of 304 and 696 on the SAT.

These tiles represent the expression x + 7x + 2.
1
Х
X X X X X X X
1
Which expression is equivalent to x + 7x + 2?
8x + 2
8x + 9
10x
9x + 1

What's the two numbers that round to the greatest place value have an estimated sum of 11000 and an estimated difference of 3000,then find the exact sum and the exact difference provide a detailed explanation as to how you reached your answers and why they will work

lucy’s hair grew 5/12 foot over the past year

a frame holds a picture that is 15 inches long and x inches wide. The frame border is 3 inches wide around the picture. what expression represents the are of the frame border

A diver went 3 feet below the surface of the water and tagged a rock with a flag, and then went down an additional 2 feet to tag another rock.
Which number line is a correct representation of the distance between the two flags?
A.
B.
C.
D.

Please help I’ll mark you as brainliest if correct

the product of two number is 20 and the sum of square is 41 find the number​

A car travels 120 miles on 2 gallons of gas. How far does the car travel on 1 gallon of gas? *

-9, -6, -1, 6, 15 …
What is the 8th term of the sequence?

Evaluate: (2.3)2

I need help fast !!!!!!!!!!!!!!!!

How to Simplify (7x+2)+(x+8)-2x

You paid $32.00 for 8 gallons of gasoline. How much would you pay for 10
gallons of gasoline?

Please help me guys

2+2

3(x+y)=y
If (x,y) is a solution to the equation above and y does not equal 0, what is the ratio x/y?

please add an explanation.

Students were asked to choose their favorite sport
and 5/9chose either basketball or soccer. If 1/9 of the
students chose basketball, what fraction chose
soccer?

The population of a town is 12,000, and it grows at a rate of 5% per year. Write an exponential function in the form [tex]y=ab^{x}[/tex] that represents the situation

The spring shown below is sitting on a flat surface and it is pushed together with a weight.




What type of energy does the spring currently have?
A.
elastic potential energy
B.
kinetic energy
C.
gravitational potential energy
D.
chemical potential energy
PLEASE I NEED HELP RIGHT NOW PLZZZ!!!

An equilateral triangle has a side length of 12sqrt(3) inches. What is the length of the altitude of the triangle?
HELP ASAP

The table below represents Mark's assets and liabilities.
Item
Value
Pocket Money
$18
Checking Account Balance $45
Loan from friend
$14
Check for mowing lawn $25
What is the total of Mark's liabilities?
O $39
O $25
O $18
O $14

In hopes of encouraging healthier snacks at school, Josie brought in a tray of carrot sticks and apple slices to share. The tray had a total of 60 snacks, of which 70% were carrot sticks. How many carrot sticks did Josie bring?

solve please!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
|−9.8|
|11 2/ 3|
|−2 6/ 7|

Can someone who is smart in math help please algebra one

Leslie leaves her house and drives to another city for her
sister's wedding. She drives 77 miles north and then 420
miles east. How far is Leslie from her house? HELP PLEASE

help pls due today thanks

Jack is shopping. Last week he bought $10 of groceries and ended up paying $10.07 after taxes. This week he's buying $15 of groceries. If the tax rate remains the same, how much will he end up paying after taxes?

Find CD if C and D are midpoints segment XY and ZY respectively.
Options:8,10,12,6

1.
Write the ratio 2:6 in its simplest form.

Which table represents the statement "Di runs at a rate of 4 meters per second " ?​

A chameleon is looking for prey. Let positive numbers represent the elevation of prey above the chameleon and negative numbers represent the elevation of prey below the chameleon. The chameleon spots a fly at 4 \text{ m}4 m4, start text, space, m, end text and a grasshopper at -6\text{ m}−6 mminus, 6, start text, space, m, end text. What does an elevation of 0 \text{ m}0 m0, start text, space, m, end text represent in this situation?

How many 1/3 are in 2 2/3