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

zequals=minus−3.5

and

zequals=3.5

​(or within 3.5 standard deviations of the​ mean).

About

nothing​%

of the area is between

zequals=minus−3.5

and

zequals=3.5

​(or within 3.5 standard deviations of the​ mean).

Answer 1

Solution :

For the given standard normal distribution, we will use the Z score.

To find the area between z= -3.5 and z= 3.5, we need to find P(-3.5<Z<3.5).

Now, P(-3.5<Z<3.5) = P(Z<3.5) - P(Z<-3.5)

Again P(Z<-3.5)= 1- P(Z<3.5)

From the standard normal table, we have P(Z<3.5)= 0.9998

Thus P(Z<-3.5)= 1- 0.9998= 0.0002

Hence P(-3.5<Z<3.5)= P(Z<3.5) - 0.0002

= 0.9998- 0.0002

= 0.9996

Again, converting this area into percent we get 99.96%.

Thus, about 99.96% of the area is between z= -3.5 and z= 3.5

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