Question

The scores on a college entrance exam have an approximate normal distribution with mean, $\mu =52$points and a standard deviation, $\sigma =11$points.

a. About $68\%$of the $y$ values lie between what two values? These values are ________________. The $z$-scores are ________________, respectively.

b. About $95\%$of the $y$values lie between what two values? These values are ________________. The $z$-scores are ________________, respectively.

c. About $99.7\%$ of the $y$ values lie between what two values? These values are ________________. The$z-$scores are ________________, respectively.

Short answer

a). The $68\%$values will fall between $41$and $63$.

b). The $95\%$values will fall between $30$and $74$.

c). The $99.7\%$values will fall between $19$and $85$.

Step-by-step solution

Given Information (Part a)

The scores on a college entrance exam have an approximate normal distribution with mean, $\mu =52$ points and a standard deviation, $\sigma =11$ points.

Explanation (Part a)

From empirical rule it is known that $68\%$will have $-1$and $1z$-scores. And, the values will be:

$x_1=\mu +z\sigma$

$=52+1\times 11$

$=63$

And,

$x_1=\mu +z\sigma$

$=52-1\times 11$

localid="1648294753461" $=41$

Therefore, the $68\%$values will fall between $\mathbf{41}$and$\mathbf{63}$.

Given Information (Part b)

The scores on a college entrance exam have an approximate normal distribution with mean,$\mu =52$ points and a standard deviation, $\sigma =11$ points.

Explanation (Part b)

From empirical rule it is known that $95\%$will have $-2$and $2z$-scores. And, the values will be:

$x_1=\mu +z\sigma$

localid="1648296180774" $=52+2\times 11$

$=74$

$x_1=\mu +z\sigma$

$=52-2\times 11$

$30$

Therefore, the $95\%$values will fall between $\mathbf{30}$and $\mathbf{74}$.

Given Information (Part c)

The scores on a college entrance exam have an approximately normal distribution with mean, $\mu =52$points and a standard deviation, $\sigma =11$ points.

Explanation

From empirical rule it is known that $99.7\%$ will have $-3$ and $3$-scores. And, the values will be:

$x_1=\mu +z\sigma$

$=52+3\times 11$

$=85$

And

$x_1=\mu +z\sigma$

$=52-3\times 11$

$=19$

Therefore, the $99.7\%$values will fall between $19$and $85$.