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
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Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
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 guidance1
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.
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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.
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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.
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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.
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Related Practice
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Describe the inflection points on the graph of a normal distribution. At what x-values are the inflection points located?
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Textbook Question
Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.
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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,
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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
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Textbook Question
The initial pressures for bicycle tires when first filled are normally distributed, with a mean of 70 pounds per square inch (psi) and a standard deviation of 1.2 psi.
b. A random sample of 15 tires is drawn from this population. What is the probability that the mean tire pressure of the sample is less than 69 psi?
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