Textbook Question
In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z = 0.72
32
views
6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:03 minutesProblem 5.Q.1b
Textbook Question
Find each probability using the standard normal distribution.
b. P(z < 2.23)
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z is less than 2.23, denoted as P(z < 2.23). The standard normal distribution has a mean of 0 and a standard deviation of 1.
Step 2: Use the standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to z = 2.23. The z-table provides the area under the curve to the left of a given z-value.
Step 3: Locate the row in the z-table corresponding to the first two digits of the z-value (2.2 in this case) and the column corresponding to the second decimal place (0.03 for 2.23).
Step 4: Read the value from the z-table at the intersection of the row and column identified in Step 3. This value represents P(z < 2.23), the cumulative probability.
Step 5: If using statistical software or a calculator, input the z-value (2.23) into the appropriate function (e.g., normalcdf in a graphing calculator or a similar function in software) to directly obtain the cumulative probability.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mWas 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 case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it essential for calculating probabilities related to normally distributed data.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table
Z-Score
A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. 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 indicates how far and in what direction a data point deviates from the mean, allowing for the calculation of probabilities.
Recommended video:
Guided course
06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) of a random variable gives the probability that the variable will take a value less than or equal to a specific value. For the standard normal distribution, the CDF can be used to find probabilities like P(z < 2.23) by looking up the z-score in standard normal distribution tables or using statistical software. This function is crucial for determining probabilities in various statistical analyses.
Recommended video:
Guided course
06:38Intro to Frequency Distributions
9:47mWatch next
Master Finding Standard Normal Probabilities using z-Table with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice