Skip to main content
Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.3.27
Chapter 5, Problem 5.3.27

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.


Find the z-score that has 2.275% of the distribution’s area to its left.

Verified step by step guidance
1
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 2.275% (or 0.02275 as a decimal) of the distribution's area lies to its left.
Step 2: Convert the given percentage to a cumulative probability. Since the area to the left of the z-score is given as 2.275%, we interpret this as the cumulative probability P(Z < z) = 0.02275.
Step 3: Use a z-table or statistical software to find the z-score corresponding to the cumulative probability of 0.02275. In a z-table, locate the value closest to 0.02275 in the body of the table and identify the corresponding z-score from the row and column headers.
Step 4: If using statistical software or a calculator, use the inverse cumulative distribution function (often denoted as invNorm or similar) to find the z-score. Input the cumulative probability of 0.02275 to obtain the z-score.
Step 5: Interpret the result. The z-score you find will be negative because 2.275% of the area is in the left tail of the standard normal distribution, which is below the mean.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key 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 means 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 represented by the z-distribution, which allows for the calculation of probabilities and z-scores. The area under the curve of the standard normal distribution corresponds to probabilities, making it a fundamental tool in statistics for determining how likely a particular z-score is.
Recommended video:
Guided course
09:47
Finding 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 z-scores, this area can be found using z-tables or statistical software. When given a specific area, such as 2.275%, one can determine the corresponding z-score that marks that percentile in the distribution, which is crucial for hypothesis testing and confidence intervals.
Recommended video:
Guided course
08:50
Z-Scores from Probabilities
Related Practice
Textbook Question

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=20, p=0.65, q=0.35

Textbook Question

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x≥109)


a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)

Textbook Question

In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.


An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.


Textbook Question

Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.


" style="" width="282">

Textbook Question

Using and Interpreting Concepts

Finding Area In Exercises 17–22, find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area.