Question

Find the area under the standard normal distribution curve for the following intervals. a. Between z = 0 and z = 2.0 b. To the right of z = 1.5 c. To the left of z = −1.75 d. Between z = −2.78 and z = 1.66

Answer 1

Answer:

a) $P(0 <z<2.0) = P(Z<2.0)-P(Z<0)$

And we can use the following excel code to find the probability:

"=NORM.DIST(2,0,1,TRUE)-NORM.DIST(0,0,1,TRUE)"

$P(0 <z<2.0) = P(Z<2.0)-P(Z<0)=0.47725$

b) $P(Z>1.5) =1-P(Z<1.5)$

And we can use the following code and we got:

"=1-NORM.DIST(1.5,0,1,TRUE)"

$P(Z>1.5) =1-P(Z<1.5)=0.0668$

c) $P(z<-1.75)$

And we can use the following code and we got:

"=NORM.DIST(-1.75,0,1,TRUE)"

$P(z<-1.75)=0.0400$

d) $P(-2.78 <z<1.66) = P(Z<1.66)-P(Z<-2.78)$

And we can use the following excel code to find the probability:

"=NORM.DIST(1.66,0,1,TRUE)-NORM.DIST(-2.78,0,1,TRUE)"

$P(-2.78 <z<1.66) = P(Z<1.66)-P(Z<-2.78)=0.949$

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".  

Solution to the problem

Part a

We want this probability:

$P(0 <z<2.0) = P(Z<2.0)-P(Z<0)$

And we can use the following excel code to find the probability:

"=NORM.DIST(2,0,1,TRUE)-NORM.DIST(0,0,1,TRUE)"

$P(0 <z<2.0) = P(Z<2.0)-P(Z<0)=0.47725$

Part b

For this case we want this probability:

$P(Z>1.5)$

And we can use the complement rule and we have:

$P(Z>1.5) =1-P(Z<1.5)$

And we can use the following code and we got:

"=1-NORM.DIST(1.5,0,1,TRUE)"

$P(Z>1.5) =1-P(Z<1.5)=0.0668$

Part c

We want this probability:

$P(z<-1.75)$

And we can use the following code and we got:

"=NORM.DIST(-1.75,0,1,TRUE)"

$P(z<-1.75)=0.0400$

Part d

We want this probability:

$P(-2.78 <z<1.66) = P(Z<1.66)-P(Z<-2.78)$

And we can use the following excel code to find the probability:

"=NORM.DIST(1.66,0,1,TRUE)-NORM.DIST(-2.78,0,1,TRUE)"

$P(-2.78 <z<1.66) = P(Z<1.66)-P(Z<-2.78)=0.949$

Answer 2

The area under the standard normal distribution curve for the interval between z = 0 and z = 2.0 is; 0.47725

How to use the normal distribution table?

A) P(0 < z < 2)

From online p-value from two z-score calculator, we have;

p-value = (0.97725 - 0.5000) = 0.47725

B) P(z > 1.5)

From online p-value from z-score calculator, we have;

p-value = 0.0668

C) P(z < 1.75)

From online p-value from z-score calculator, we have;

p-value = 1 - 0.040059

p-value = 0.9599

D) P(-2.78 < z < 1.6)

From online p-value from two z-score calculator, we have;

p-value = 0.94882

Read more about Normal Distribution table at; https://brainly.com/question/4079902

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