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 5.3.30
Chapter 5, Problem 5.3.30

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


Find the positive z-score for which 12% of the distribution’s area lies between and z.

Verified step by step guidance
1
Step 1: Understand the problem. You are tasked with finding the positive z-score such that 12% (or 0.12) of the area under the standard normal distribution curve lies between 0 and the z-score. This means the area to the left of the z-score is 0.5 (from the left of the mean to 0) plus 0.12.
Step 2: Calculate the cumulative area to the left of the z-score. Add the area from the mean to the z-score (0.12) to the area from the far left of the curve to the mean (0.5). This gives a cumulative area of 0.5 + 0.12 = 0.62.
Step 3: Use a z-table or statistical software to find the z-score corresponding to a cumulative area of 0.62. A z-table provides the cumulative probability for a given z-score, so you will look for the value closest to 0.62 in the table.
Step 4: Identify the z-score from the z-table or software. Locate the row and column in the z-table that correspond to the cumulative probability of 0.62. The intersection of the row and column gives the z-score.
Step 5: Verify your result. Ensure that the z-score you found is positive and that the cumulative area to the left of this z-score matches 0.62. This confirms that 12% of the area lies between the mean and the z-score.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
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-table, which provides the area (probability) to the left of a given z-score. Understanding this distribution is crucial for finding probabilities and z-scores, as it allows for the conversion of any normal distribution into a standard form for easier analysis.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table

Area Under the Curve

In statistics, the area under the curve (AUC) 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 used to find probabilities associated with z-scores. In the context of the question, finding the z-score for which a specific area (12% in this case) lies between the mean and that z-score involves understanding how to interpret and calculate areas under the normal curve.
Recommended video:
Guided course
08:50
Z-Scores from Probabilities
Related Practice
Textbook Question

Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results.


Asthma Prevalence by State The mean percent of asthma prevalence of the 50 U.S. states is 9.51%. A random sample of 30 states is selected. What is the probability that the mean percent of asthma prevalence for the sample is greater than 10%? Assume sigma=1.17%


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.

Between z= -1.55 and z= 1.55

Textbook Question

Describe the inflection points on the graph of a normal distribution. At what x-values are the inflection points located?

Textbook Question

Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (a) fewer than four personal fouls,

Textbook Question

In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.


Mu = 790, sigma =48, n = 250

Textbook Question

A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup)


c. Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning.