Question

Suppose that the national average for the math portion of the College Board's SAT is 510. The College Board periodically rescales the test scores such that the standard deviation is approximately 100. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. If required, round your answers to two decimal places. If your answer is negative use "minus sign". (a) What percentage of students have an SAT math score greater than 610? % (b) What percentage of students have an SAT math score greater than 710? % (c) What percentage of students have an SAT math score between 410 and 510? % (d) What is the z-score for student with an SAT math score of 625? (e) What is the z-score for a student with an SAT math score of 415?

Answer 1

Answer:

a) 0.1587 b) 0.023 c) 0.1587 d) 1.15 e)-0.95      

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 510

Standard Deviation, σ = 100

We are given that the distribution of SAT score is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(score greater than 610)

P(x > 610)

$P( x > 610) = P( z > \displaystyle\frac{610 - 510}{100}) = P(z > 1)$

$= 1 - P(z \leq 1)$

Calculation the value from standard normal z table, we have,

$P(x > 610) = 1 - 0.8413 = 0.1587 = 15.87\%$

b) P(score greater than 710)

$P(x > 710) = P(z > \displaystyle\frac{710-510}{100}) = P(z > 2)\\\\P( z > 2) = 1 - P(z \leq 2)$

Calculating the value from the standard normal table we have,

$1 - 0.977 = 0.023 = 2.3\%\\P( x > 710) = 2.3\%$

c)P(score between 410 and 510)

$P(410 \leq x \leq 510) = P(\displaystyle\frac{410 - 510}{100} \leq z \leq \displaystyle\frac{510-510}{100}) = P(-1 \leq z \leq 0)\\\\= P(z \leq 0) - P(z < -1)\\= 0.500 - 0.159 = 0.341 = 34.1\%$

$P(410 \leq x \leq 510) = 34.1\%$

d) x = 625

$z_{score} = \displaystyle\frac{625 - 510}{100} = \displaystyle\frac{115}{100} = 1.15$

e) x = 415

$z_{score} = \displaystyle\frac{415 - 510}{100} = \displaystyle\frac{-95}{100} = -0.95$