Find the probabilities using the normal distribution.

a. Between [tex]$z=0$[/tex] and [tex]$z=1.97$[/tex]

b. Between [tex][tex]$z=0.79$[/tex][/tex] and [tex]$z=1.28$[/tex]

c. To the left of [tex]$z=1.22$[/tex]

d. To the right of [tex][tex]$z=-1.9$[/tex][/tex]

e. To the left of [tex]$z=-2.15$[/tex] or to the right of [tex]$z=1.6$[/tex]

f. [tex][tex]$P(z\ \textgreater \ 2.83)$[/tex][/tex]

g. [tex]$P(-0.05\ \textless \ z\ \textless \ 1.10)$[/tex]



Answer :

Sure, let's break down each part of the problem step-by-step:

### Part a: Between [tex]\( z = 0 \)[/tex] and [tex]\( z = 1.97 \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] falls between 0 and 1.97. This is calculated as the difference between the cumulative distribution function (CDF) values at these two points:

[tex]\[ P(0 \leq Z \leq 1.97) = \Phi(1.97) - \Phi(0) \][/tex]

Given the numerical result:

[tex]\[ P(0 \leq Z \leq 1.97) = 0.4755808147197774 \][/tex]

### Part b: Between [tex]\( z = 0.79 \)[/tex] and [tex]\( z = 1.28 \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] falls between 0.79 and 1.28:

[tex]\[ P(0.79 \leq Z \leq 1.28) = \Phi(1.28) - \Phi(0.79) \][/tex]

Given the numerical result:

[tex]\[ P(0.79 \leq Z \leq 1.28) = 0.11449131620919506 \][/tex]

### Part c: To the left of [tex]\( z = 1.22 \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] is less than 1.22:

[tex]\[ P(Z \leq 1.22) = \Phi(1.22) \][/tex]

Given the numerical result:

[tex]\[ P(Z \leq 1.22) = 0.8887675625521654 \][/tex]

### Part d: To the right of [tex]\( z = -1.9 \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] is greater than -1.9:

[tex]\[ P(Z \geq -1.9) = 1 - \Phi(-1.9) \][/tex]

Given the numerical result:

[tex]\[ P(Z \geq -1.9) = 0.9712834401839981 \][/tex]

### Part e: To the left of [tex]\( z = -2.15 \)[/tex] or to the right of [tex]\( z = 1.6 \)[/tex]

We need to find two probabilities and sum them up:

1. The probability [tex]\( Z \)[/tex] is less than -2.15:
[tex]\[ P(Z \leq -2.15) = \Phi(-2.15) \][/tex]

2. The probability [tex]\( Z \)[/tex] is greater than 1.6:
[tex]\[ P(Z \geq 1.6) = 1 - \Phi(1.6) \][/tex]

So the total probability is:
[tex]\[ P(Z \leq -2.15) + P(Z \geq 1.6) = \Phi(-2.15) + (1 - \Phi(1.6)) \][/tex]

Given the numerical result:

[tex]\[ P(Z \leq -2.15 \text{ or } Z \geq 1.6) = 0.07057689909064849 \][/tex]

### Part f: [tex]\( P(z > 2.83) \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] is greater than 2.83:

[tex]\[ P(Z \geq 2.83) = 1 - \Phi(2.83) \][/tex]

Given the numerical result:

[tex]\[ P(Z \geq 2.83) = 0.0023274002067315003 \][/tex]

### Part g: [tex]\( P(-0.05 < z < 1.10) \)[/tex]

We want to find the probability that a standard normal variable [tex]\( Z \)[/tex] falls between -0.05 and 1.10:

[tex]\[ P(-0.05 \leq Z \leq 1.10) = \Phi(1.10) - \Phi(-0.05) \][/tex]

Given the numerical result:

[tex]\[ P(-0.05 \leq Z \leq 1.10) = 0.3842727448919898 \][/tex]

So, these are the detailed step-by-step solutions for the given probabilities using the standard normal distribution.