Given a standard normal distribution, find the area under the curve t

Cynthia Bell 2021-12-20 Answered

Given a standard normal distribution, find the area under the curve that lies 
(a) to the left of z=1.39
(b) to the right of z=1.96
(c) between z=2.16  and  z=0.65
(d) to the left of z=1.43
(e) to the right of z=0.89
(f) between z=0.48  and  z=1.74.

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Expert Answer

turtletalk75
Answered 2021-12-21 Author has 29 answers

Step 1
a)
Let's find the area under the curve that lies to the left of z=1.39. So, we need to find P(Z<1.39), where Z represent Standard Normal random variable.
Using Normal Probability Table, we easily obtain:
P(Z<1.39)=0.0823
Step 2
b)
Let's now find the area the curve that lies to the right of z=1.96. So, we need to find P(Z<1.96), where Z represent Standard Normal random variable.
Using Normal Probability Table, we obtain:
P(Z>1.96)=1P(Z<1.96)
=10.9750
=0.025
Step 3
c)
Let's now find the area under the curve that lies between z=2.16  and  z=0.65.
So, we need to find P(2.16<Z<0.65), where Z represent Standard Normal random variable.
Using Normal Probability Table, we obtain:
P(2.16<Z<0.65)=P(Z<0.65)P(Z<2.16)
=0.25780.0154
=0.2424
Step 4
d)
Let's find the area under the curve that lies to the left of z=1.43. So, we need to fidn P(Z<1.43), where Z represent Standard Normal random variable.
Using Normal Probability Table, we obtain:
P(Z<1.43)=0.9236
Step 5
e)
Let's find the area under the curve that lies to the right of z=0.89. Som we need to find P(Z0.89), where Z represent Standard Normal random variable.
Using Normal Probability Table, we obtain:
P(Z0.896)=1P(Z<0.89)
=10.1867
=0.8133
Step 6
f)
Let's now find the area under the curve that lies between z=0.48  and  z=1.74.
So, we need to find P(0.48<Z<1.74), where Z represent Standard Normal random variable.
Using Normal Probability Table, we obtain:
P(0.48<Z<1.74)=P(Z<1.74)P(Z<0.48)
=0.95910.3156
=0.6435.

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Bubich13
Answered 2021-12-22 Author has 36 answers

Answer: 
a) P(Z<1.39)=0.082264 
b) P(Z>1.96)=0.025 
c) P(2.16<Z<0.65)=0.2457 
d) P(Z<1.43)=0.92364 
e) P(Z0.48)=0.68439 
f) P(0.48<Z<1.74)=0.6435 
Step-by-step explanation: 
Considering a 
From the question we are told that 
The z-score is z=1.39
Generally from the z table the area under the curve that lies to the left of z=1.39 is P(Z<1.39)=0.082264 
Considering b 
From the question we are told that 
The z-score is z=1.96 
Generally from the z table the area under the curve that lies to the right of z=1.96 is P(Z>1.96)=0.025 
Considering c 
From the question we are told that 
The z-score is z=2.16  and  z=0.65 
Generally from the z table the area under the curve that lies between z=2.16  and  z=0.65
P(2.16<Z<0.65)=P(Z<0.64)P(Z<2.16) 
From the z table the area under (Z<0.64)  and  (Z<2.16) is P(Z<0.64)=0.26109 
and P(Z<2.16)=0.015386 
So P(2.16<Z<0.65)=0.261090.015386 
P(2.16<Z<0.65)=0.2457 
Considering d 
From the question we are told that 
The z-score is z=1.45 
Generally from the z table the area under the curve that lies to the left of z=1.43 is P(Z<1.43)=0.92364 
Considering e 
From the question we are told that 
The z-score is z=0.48 
Generally from the z table the area under the curve that lies to the right of z=0.48 is P(Z0.48)=0.68439 
Considering f 
From the question we are told that 
The z-score is z=0.48  and  z=1.74 
Generally from the z table the area under the curve that lies between z=2.16  and  z=0.65
P(0.48<Z<1.74)=P(Z<1.74)P(Z<0.48) 
From the z table the area under (Z<1.74)  and  (Z<0.48) is P(Z<1.74)=0.95907 
and P(Z<0.48)=0.31561 
 

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