Let the random variable X follow a normal distribution with μ=50 and σ^2=64. a. Find the probabi

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 $\mu=50$ and $\sigma^{2}=64$. a. Find the probability that $X$ is greater than 60 . b. Find the probability that $X$ is greater than 35 and less than 62 . c. Find the probability that $X$ is less than 55 . d. The probability is $0.2$ that $X$ is greater than what number? e. The probability is $0.05$ that $X$ is in the symmetric interval about the mean between which two numbers?

Let the random variable $X$ follow a normal distribution with $\mu=50$ and $\sigma^{2}=64$.
a. Find the probability that $X$ is greater than 60 .
b. Find the probability that $X$ is greater than 35 and less than 62 .
c. Find the probability that $X$ is less than 55 .
d. The probability is $0.2$ that $X$ is greater than what number?
e. The probability is $0.05$ that $X$ is in the symmetric interval about the mean between which two numbers?

Key Concepts

Symmetry of the Normal Distribution

The normal distribution is symmetric about its mean. This symmetry implies that percentages in the tails of the distribution are equal for deviations of the same magnitude above and below the mean. This property is useful when determining symmetric confidence intervals and probability intervals around the mean.

Inverse CDF (Quantile Function)

The inverse CDF, or quantile function, is used to determine the value of a random variable corresponding to a given cumulative probability. In practice, it helps in finding threshold values, such as percentiles, where a certain percentage of the observations lie below or above these values.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) gives the probability that a random variable takes a value less than or equal to a given point. In the context of the normal distribution, it is used to calculate the probabilities associated with ranges of values by relating the variable to standard normal probabilities.

Standardization (Z-score Transformation)

Standardization is the process of converting a normally distributed variable to a standard normal distribution with a mean of 0 and a standard deviation of 1. This is done by subtracting the mean and dividing by the standard deviation. The transformation allows for the use of standard normal tables or computational tools to find probabilities for any normally distributed variable.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric around its mean. It is characterized by the mean (?) and variance (?²), and it has the familiar bell-shaped curve. This distribution is fundamental in probability and statistics, especially in the context of measurements, errors, and natural phenomena.

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