Question

Find the proportion of observations (±0.0001) from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region.
(a) z gif.latex?%5Cleqslant ?2.34:
(b) z gif.latex?%5Cgeqslant ?2.34:
(c) z > 1.74:
(d) ?2.34 < z < 1.74:

Answer 1

Answer:

a) We can use the following excel code to find it:"=NORM.DIST(-2.34,0,1,TRUE)"

$P(Z \leq -2.34)=0.0096$

b) $P(Z \geq -2.34)= 1-P(X \leq -2.34)= 1-0.0096=0.9904$

c) We can use the following excel code: "=1-NORM.DIST(1.74,0,1,TRUE)"

$P(Z > 1.74)= 1-P(X \leq 1.74)=0.0409$

d) We can use the following excel code: "=NORM.DIST(1.74,0,1,TRUE)-NORM.DIST(-2.34,0,1,TRUE)"

$P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)=0.959-0.0096=0.949$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

We want to find this probability:

$P(Z \leq -2.34)$

And we can use the following excel code to find it:"=NORM.DIST(-2.34,0,1,TRUE)"

$P(Z \leq -2.34)=0.0096$

Part b

We want to find this probability:

$P(Z \geq -2.34)$

And for this case we can use the complement rule:

$P(Z \geq -2.34)= 1-P(X \leq -2.34)$

And using the result from part a we got:

$P(Z \geq -2.34)= 1-P(X \leq -2.34)= 1-0.0096=0.9904$

Part c

We want to find this probability:

$P(Z > 1.74)$

And for this case we can use the complement rule:

$P(Z > 1.74)= 1-P(X \leq 1.74)$

And we can use the following excel code: "=1-NORM.DIST(1.74,0,1,TRUE)"

$P(Z > 1.74)= 1-P(X \leq 1.74)=0.0409$

Part d

We want to find this probability:

$P(-2.34<Z <1.74)$

And for this case we can find this probability with this difference:

$P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)$

And we can use the following excel code: "=NORM.DIST(1.74,0,1,TRUE)-NORM.DIST(-2.34,0,1,TRUE)"

$P(-2.34<Z < 1.74)= P(Z<1.74)-P(Z<-2.34)=0.959-0.0096=0.949$