Question

Find the indicated area under the curve of the standard normal distribution, then convert it to a percentage and fill in the blank.

About _____% of the area is between z=?2.2 and z=2.2 (or within 2.2 standard deviations of the mean).

About?____% of the area is between z=?2.2 and z=2.2 (or within 2.2 standard deviations of the mean).

(Round to two decimal places as needed.)?

Answer 1

Concepts and reason

The normal distribution is a continuous distribution and important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distribution is not known. The standard normal distribution is a type of normal distribution with a mean equal to zero and the standard deviation is equal to \(1 .\)

Fundamentals

The Excel formula calculating the probability value is, \(=\) NORMSDIST \((z)\)

The formula for between probability is as follows:

\(P(a \leq z \leq b)=P(z \leq b)-P(z \leq a)\)

 

From the information, observe that the standard normal \(z\) score lies between -2 and +2. The probability that the z score is less than - 2 is,

$$ P(z<-2)=0.0228 \quad(=\text { NORMSDIST }(-2)) $$

The probability that the \(z\) score is less than 2 is,

$$ P(z<2)=0.9772 \quad(=\text { NORMSDIST }(2)) $$

The calculated value of \(z\) scores less than -2 is 0.0228 The calculated value of \(z\) score less than 2 is 0.9772 These values are used to calculate the probability that the standard z score lies between -2 and +2.

The calculation of the required probability is,

$$ \begin{aligned} P(-2<z<2) &=P(z<2)-P(z<-2) \\ &=0.9772-0.0228 \\ &=0.9545 \\ & \approx 95.45 \% \end{aligned} $$

About \(95.45 \%\) of the area between \(z=-2\) and \(z=2.2\) (Or within 2 standard deviations of the mean).  It is symbolically expressed as follows: