6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:56 minutesProblem 5.1.29
Textbook Question
Finding Area
In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the right of z= -0.355
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the area under the standard normal curve to the right of z = -0.355. The standard normal curve is symmetric, with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the total area under the standard normal curve is 1. The area to the right of a given z-score can be found using the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the formula for the cumulative area to the left of z, which is given by P(Z ≤ z). For z = -0.355, find the cumulative area to the left of this z-score using a z-table or statistical software.
Step 4: Subtract the cumulative area to the left of z = -0.355 from 1 to find the area to the right. Mathematically, this is expressed as P(Z > -0.355) = 1 - P(Z ≤ -0.355).
Step 5: If using technology, input z = -0.355 into a statistical calculator or software to directly compute the area to the right. Alternatively, use a z-table to find P(Z ≤ -0.355), then subtract it from 1 as described in Step 4.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Normal Distribution
The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is used to describe how data is distributed in a standardized way, allowing for comparison across different datasets. The z-score represents the number of standard deviations a data point is from the mean, facilitating the calculation of probabilities and areas under the curve.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table
Z-Score
A z-score indicates how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In the context of the standard normal distribution, a z-score of -0.355 means the value is 0.355 standard deviations below the mean, which is essential for determining the area to the right of this z-score.
Recommended video:
Guided course
06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Area Under the Curve
The area under the curve of a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be found using z-tables or technology, such as statistical software or calculators. In this case, finding the area to the right of z = -0.355 involves calculating the probability that a value is greater than this z-score.
Recommended video:
Guided course
08:50Z-Scores from Probabilities
9:47mWatch next
Master Finding Standard Normal Probabilities using z-Table with a bite sized video explanation from Patrick
Start learning
