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
Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
Not the one you use?Change textbookSelect a different textbook
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.CR.9
In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
Between z = 0 and z = 2.95
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve between z = 0 and z = 2.95. The standard normal curve is a bell-shaped curve with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the area under the standard normal curve represents probabilities. To find the area between two z-scores, you can use the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the formula for the cumulative area under the curve: \( P(a < Z < b) = P(Z < b) - P(Z < a) \). Here, \( a = 0 \) and \( b = 2.95 \). This means you need to calculate \( P(Z < 2.95) \) and \( P(Z < 0) \).
Step 4: Use a z-table or technology (such as a statistical calculator or software) to find the cumulative probabilities. From the z-table or technology, find \( P(Z < 2.95) \), which gives the cumulative area to the left of z = 2.95, and \( P(Z < 0) \), which gives the cumulative area to the left of z = 0.
Step 5: Subtract \( P(Z < 0) \) from \( P(Z < 2.95) \) to find the area between z = 0 and z = 2.95. This difference represents the desired area under the standard normal curve.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas 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 normal distribution with a mean of 0 and a standard deviation of 1. It is represented by the z-score, which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve, as it allows for the standardization of different normal distributions.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table
Z-scores
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. Z-scores are essential for determining the area under the standard normal curve, as they allow us to find probabilities associated with specific values.
Recommended video:
Guided course
06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Area Under the Curve
The area under the curve in a standard normal distribution represents the probability of a random variable falling within a particular range. To find the area between two z-scores, one can use statistical tables or technology, such as calculators or software, which provide the cumulative probabilities associated with those z-scores. This area is critical for making inferences about data and understanding the likelihood of certain outcomes.
Recommended video:
Guided course
08:50Z-Scores from Probabilities
Related Practice
Textbook Question
Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (c) less than two. (d) Are any of these events unusual? Explain your reasoning.
1
views
Textbook Question
Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (a) exactly 12,
1
views
Textbook Question
Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (b) between 8 and 11, inclusive,
1
views
Textbook Question
The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months.
c. What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?
Textbook Question
Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (c) between two and four personal fouls, inclusive.