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Sagot :
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.
### 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.
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