Let the random variable X follow a normal distribution with μ=80 and σ^2=100. a. Find the probab

step 1 Now here on this problem, we are told that X is a random variable following a normal distributio

Let the random variable X follow a normal distribution with...

Let the random variable $X$ follow a normal distribution with $\mu=80$ and $\sigma^{2}=100$. a. Find the probability that $X$ is greater than 60 . b. Find the probability that $X$ is greater than 72 and less than 82 . c. Find the probability that $X$ is less than $55 .$ d. The probability is $0.1$ that $X$ is greater than what number? e. The probability is $0.6826$ that $X$ is in the symmetric interval about the mean between which two numbers?

Let the random variable $X$ follow a normal distribution with $\mu=80$ and $\sigma^{2}=100$.
a. Find the probability that $X$ is greater than 60 .
b. Find the probability that $X$ is greater than 72 and less than 82 .
c. Find the probability that $X$ is less than $55 .$
d. The probability is $0.1$ that $X$ is greater than what number?
e. The probability is $0.6826$ that $X$ is in the symmetric interval about the mean between which two numbers?

Key Concepts

Symmetric Interval about the Mean

An interval centered at the mean of a distribution that extends equally in both directions. In the case of a normal distribution, these intervals are used to capture a specified proportion of the total probability, taking advantage of the distribution’s symmetry to define bounds around the mean.

Percentiles and Quantiles

Values that divide a distribution into portions with equal probabilities. They are used to determine specific threshold values corresponding to given cumulative probabilities, which is useful in tasks such as identifying cutoff points or constructing confidence intervals in a normal distribution.

Cumulative Distribution Function

A function that provides the probability that a random variable will be less than or equal to a given value. In the context of a normal distribution, it is used to find the area under the probability curve up to a specified point, allowing for the computation of probabilities over intervals.

Z-Score Transformation

A process that converts a normal random variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This is essential for comparing different normal distributions and for using standard normal tables or functions to determine probabilities.

Normal Distribution

A probability distribution that is symmetric and bell-shaped, characterized by its mean and variance. It is widely used in statistics to model real-world phenomena where data tends to cluster around a central value, with probabilities computed by measuring the area under its curve.

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