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Answer :
Certainly! Let's solve the problem step-by-step.
Question: What is the probability that a variable following a standard normal distribution falls between 1.70 and 2.40?
Solution:
To solve this, we use the properties of the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. We are interested in finding the probability that a random variable [tex]\( Z \)[/tex] falls between 1.70 and 2.40.
1. Identify the Bounds:
- Lower bound: 1.70
- Upper bound: 2.40
2. Use the Cumulative Distribution Function (CDF):
The standard normal cumulative distribution function can be used to find the probability that [tex]\( Z \)[/tex] is less than a certain value. Let's denote:
- [tex]\( P(Z < 1.70) = 0.0455 \)[/tex]
- [tex]\( P(Z < 2.40) = 0.9918 \)[/tex]
3. Find the Desired Probability:
To find the probability that [tex]\( Z \)[/tex] is between 1.70 and 2.40, subtract the cumulative probability up to 1.70 from the cumulative probability up to 2.40:
[tex]\[
P(1.70 < Z < 2.40) = P(Z < 2.40) - P(Z < 1.70) = 0.9918 - 0.0455 = 0.9463
\][/tex]
Thus, the probability that the variable falls between 1.70 and 2.40 in a standard normal distribution is [tex]\( 94.63\% \)[/tex].
Question: What is the probability that a variable following a standard normal distribution falls between 1.70 and 2.40?
Solution:
To solve this, we use the properties of the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. We are interested in finding the probability that a random variable [tex]\( Z \)[/tex] falls between 1.70 and 2.40.
1. Identify the Bounds:
- Lower bound: 1.70
- Upper bound: 2.40
2. Use the Cumulative Distribution Function (CDF):
The standard normal cumulative distribution function can be used to find the probability that [tex]\( Z \)[/tex] is less than a certain value. Let's denote:
- [tex]\( P(Z < 1.70) = 0.0455 \)[/tex]
- [tex]\( P(Z < 2.40) = 0.9918 \)[/tex]
3. Find the Desired Probability:
To find the probability that [tex]\( Z \)[/tex] is between 1.70 and 2.40, subtract the cumulative probability up to 1.70 from the cumulative probability up to 2.40:
[tex]\[
P(1.70 < Z < 2.40) = P(Z < 2.40) - P(Z < 1.70) = 0.9918 - 0.0455 = 0.9463
\][/tex]
Thus, the probability that the variable falls between 1.70 and 2.40 in a standard normal distribution is [tex]\( 94.63\% \)[/tex].
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