Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. z = (x-μ)/σ, for x (standardized value)

Verified step by step guidance

Start with the given formula: \(z = \frac{x - \mu}{\sigma}\).

To solve for \(x\), multiply both sides of the equation by \(\sigma\) to eliminate the denominator: \(z \times \sigma = x - \mu\).

Next, isolate \(x\) by adding \(\mu\) to both sides: \(x = z \times \sigma + \mu\).

This expression now represents \(x\) in terms of \(z\), \(\mu\), and \(\sigma\).

Remember that \(\sigma\) (standard deviation) is assumed to be nonzero to avoid division by zero.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Formulas for a Specific Variable

This involves rearranging an equation to isolate the desired variable on one side. It requires applying inverse operations such as addition, subtraction, multiplication, division, and sometimes more advanced algebraic techniques to rewrite the formula clearly in terms of the specified variable.

Understanding the Standardization Formula

The formula z = (x - μ) / σ represents the standard score or z-score, which measures how many standard deviations a data point x is from the mean μ. Recognizing the roles of each variable helps in correctly manipulating the formula to solve for x.

Handling Variables in the Denominator

When variables appear in the denominator, it is important to assume they are nonzero to avoid undefined expressions. This assumption allows safe multiplication across the equation to eliminate fractions and solve for the desired variable without division by zero errors.

Recommended video:

Guided course

02:58

Rationalizing Denominators

Related Practice

Textbook Question

Height of a Projected Ball An astronaut on the moon throws a baseball upward. The astronaut is 6 ft, 6 in. tall, and the initial velocity of the ball is 30 ft per sec. The height s of the ball in feet is given by the equations=-2.7t²+30t+6.5,where t is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 ft above the moon's surface? Round to the nearest hundredth. (b) How many seconds will it take for the ball to hit the moon's surface? Round to the nearest hundredth.

Textbook Question

Solve each quadratic inequality. Give the solution set in interval notation. x(x+1)<12

Textbook Question

Solve each equation or inequality. | 5x + 2 | - 2 < 3

Textbook Question

Solve each equation using completing the square. 3x² - 9x + 7 = 0

Textbook Question

Solve each equation. √(4x+13) = 2x-1

Textbook Question

Find each sum or difference. Write answers in standard form. (3+2i) + (9+3i)