91Ó°ÊÓ

Let \(x\) be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of 75 . a. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is \(.0250\). b. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is \(.9345\). c. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is approximately \(.0275 .\) d. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is approximately . 9600 . e. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4700\) and the value of \(x\) is less than \(\mu .\) f. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4100\) and the value of \(x\) is greater than \(\mu\).

Step by step solution

01

Identify values

Identify the known values. In this case, the mean (\( \mu \)) is 550 and the standard deviation (\( \sigma \)) is 75.

02

Use standard normal distribution table

Use the standard normal distribution table (z-table) to find the z-score corresponding to the given probability. The z-score corresponds to the number of standard deviations away from the mean. For instance, in part a, a probability of .0250 corresponds approximately to a z-score of -1.96, found in the standard normal distribution table.

03

Apply z-score formula

Apply the z-score formula, which is \( Z = \frac{x - \mu}{\sigma} \), to find the unknown 'x' for each part of this exercise. Solve for 'x' in the equation.

04

Repeat for each part

Repeat the process for each part of the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Random Variable

A continuous random variable is a type of variable that can take on an infinite number of possible values within a given range. Unlike discrete variables, which have specific set values, continuous variables can represent any value within a continuous range. In the context of a normal distribution, a continuous random variable like our variable "x" can assume a vast spectrum of values between any two points.

• Continuous random variables are often used in modeling natural phenomena, like the height of people, temperature, or time.
• The probability of a continuous random variable taking on any specific value is theoretically zero. Instead, probabilities are measured over intervals.
• A common function used to describe continuous random variables is the probability density function (PDF).

In our exercise, the variable "x" is continuous, relying on a probability distribution to determine ranges of outcomes.

Z-score

The z-score is a crucial statistical measurement that represents how many standard deviations an element is from the mean of its data set. When dealing with normal distributions, z-scores allow us to standardize and transform random variables.

• The z-score is given by the formula: \[ Z = \frac{x - \mu}{\sigma} \]where \( x \) is the value in question, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• A positive z-score indicates the value is above the mean, while a negative z-score shows it is below the mean.
• Z-scores are essential for determining probabilities and percentiles within a normal distribution.

Understanding z-scores is fundamental when using the normal distribution to find the probability of variables like "x" in our original exercise.

Standard Normal Distribution Table

The standard normal distribution table, commonly known as the z-table, helps in finding the probabilities of a standard normal distribution. It assists in converting a normal distribution with any mean and standard deviation to a standard distribution, which has a mean of 0 and a standard deviation of 1.

• The table lists z-scores and their corresponding probabilities under the curve to its left.
• To use the z-table, first calculate the z-score using the formula mentioned earlier. Then, find this z-score in the table to see the probability associated with it.
• These probabilities allow us to find the areas under the curve efficiently, which are fundamental in deriving the probability or determining the cutoff values in a normal distribution.

Using the standard normal distribution table is a vital step in transforming data into a more analyzable form, as shown in the step-by-step solution.

Mean and Standard Deviation

The mean and standard deviation are critical components of any normal distribution. They help in defining the curve and deciding the spread and center point of the distribution.

• The mean, \( \mu \), is the average of all values. It determines the central point of the distribution.
• Standard deviation, \( \sigma \), measures the dispersion of a dataset relative to its mean. It indicates how spread out the values are around the mean.
• Together, these values shape the normal distribution curve, where approximately 68% of values lie within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean.

In our exercise, the mean of 550 and a standard deviation of 75 define our normal distribution, allowing us to calculate z-scores and look for probabilities using the z-table.