Question

A certain standardized test's math scores have a bell-shaped distribution with a mean of 530 and a standard deviation of 119. Complete parts (a) through (c) (a) What percentage of standardized test scores is between 411 and 649? 68% (Round to one decimal place as needed.) (b) What percentage of standardized test scores is less than 411 or greater than 649? 1 32% (Round to one decimal place as needed.) (c) What percentage of standardized test scores is greater than 768? % (Round to one decimal place as needed.)

Answer 1

Answer:

a) 68.2%

b) 31.8%

c) 2.3%

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 530

Standard Deviation, σ = 119

We are given that the distribution of math scores is a bell shaped distribution that is a normal distribution.

Formula:

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

a) P(test scores is between 411 and 649)

$P(411 \leq x \leq 649)\\\\= P(\displaystyle\frac{411 - 530}{119} \leq z \leq \displaystyle\frac{649-530}{119})\\\\= P(-1 \leq z \leq 1)\\\\= P(z \leq 1) - P(z < -1)\\= 0.841 - 0.159 = 0.682 = 68.2\%$

b) P(scores is less than 411 or greater than 649)

$P(x < 411 < x, x > 649)\\=1 = P(411 \leq x \leq 649)\\=1 - 0.682\\=0.318 = 31.8\%$

c) P(score greater than 768)

P(x > 768)

$P( x > 768) = P( z > \displaystyle\frac{768 - 530}{119}) = P(z > 2)$

$= 1 - P(z \leq 2)$

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

$P(x > 768) = 1 -0.977 = 0.023 = 2.3\%$