Textbook Question
Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.
P1.5
95
views
6. Normal Distribution and Continuous Random Variables
Probabilities & Z-Scores w/ Graphing Calculator
1:41 minutesProblem 5.3.24
Textbook Question
Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.
Find the z-score that has 78.5% of the distribution’s area to its left.
Verified step by step guidance1
Step 1: Understand the problem. The z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution. Here, we are tasked with finding the z-score such that 78.5% of the distribution's area lies to its left.
Step 2: Recall that the cumulative area to the left of a z-score in a standard normal distribution can be found using a z-table, statistical software, or a calculator with statistical functions. The cumulative area corresponds to the probability given in the problem, which is 0.785.
Step 3: Use the z-table or statistical software to find the z-score that corresponds to a cumulative probability of 0.785. In a z-table, locate the value closest to 0.785 in the body of the table, and then identify the corresponding z-score from the row and column headers.
Step 4: If using a calculator or statistical software, use the inverse cumulative distribution function (often denoted as invNorm or similar) to input the cumulative probability of 0.785 and obtain the z-score.
Step 5: Interpret the result. The z-score you find will indicate how many standard deviations above or below the mean the point is, such that 78.5% of the distribution's area lies to its left.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Z-Score
A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below. Z-scores are essential for standardizing scores on different scales and for comparing data points from different distributions.
Recommended video:
Guided course
06:31
Z-Scores From Given Probability - TI-84 (CE) Calculator
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 symmetrical and bell-shaped, allowing for the use of z-scores to find probabilities and percentiles. The area under the curve represents the total probability, and specific z-scores correspond to specific areas, making it crucial for determining probabilities in statistics.
Recommended video:
Guided course
09:47Finding Standard Normal Probabilities using z-Table
Area Under the Curve
In the context of the normal distribution, the area under the curve represents the probability of a random variable falling within a particular range. For a given z-score, the area to the left indicates the proportion of the distribution that is less than that z-score. This concept is vital for finding z-scores corresponding to specific probabilities, such as the 78.5% area mentioned in the question.
Recommended video:
Guided course
08:50Z-Scores from Probabilities
Related Videos
Related Practice