The scores on a standardized graduate admissions test are normally distributed with a mean… — GRE General Test

The scores on a standardized graduate admissions test are normally distributed with a mean of $70$ and a standard deviation of $10$ . Which of the following statements must be true? Select all such statements.

A test score of $80$ corresponds to a $z$ -score of $+1$ .Answer
Approximately $68\%$ of all test scores fall between $60$ and $80$ .Answer
C
A test score of $70$ corresponds to a $z$ -score of $70$ .
D
A test score of $50$ is $1$ standard deviation above the mean.
E
A test score of $70$ is at approximately the $95\text{th}$ percentile.

Answer

The correct statements are that a score of 80 corresponds to a z-score of +1 and that approximately 68% of all test scores fall between 60 and 80.

The statement asserting that a test score of

80

has a

z

-score of

+1

is true because

80

is exactly

1

standard deviation (

10

units) greater than the mean of

70

. The statement that approximately

68\%

of test scores fall between

60

and

80

is also true according to the empirical rule, which dictates that approximately

68\%

of values in a normal distribution lie within one standard deviation of the mean (

[70-10, 70+10]

Step-by-Step Solution

Calculate the z-score for a score of 80

z = \frac{80 - 70}{10} = +1

The

z

-score formula is

z = \frac{X - \mu}{\sigma}

, where

X

is the data value,

\mu

is the mean, and

\sigma

is the standard deviation.

Apply the Empirical Rule (68-95-99.7 Rule) for 1 standard deviation

Interval

[70 - 10, 70 + 10] = [60, 80]

contains approximately

68\%

of the distribution

In any normal distribution, about

68\%

of observations lie within

\mu \pm 1\sigma

Evaluate the symmetry and percentile rank at the mean

Score of 70 is at the

50\text{th}

percentile and has a

z

-score of

0

The mean of a normal distribution is equal to its median, bisecting the area under the curve into two equal halves of

50\%

Key Concept

Empirical Rule and z-score calculation in a Normal Distribution

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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