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.
To the left of z=1.365
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 5.3.24
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.
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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 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.
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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 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.
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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.
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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=18, p=0.90, q=0.10
Textbook Question
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Textbook Question
Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.
Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.
Textbook Question
Bags of Baby Carrots The weights of bags of baby carrots are normally distributed, with a mean of 32 ounces and a standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the most a bag of baby carrots can weigh and not need to be repackaged?
Textbook Question
True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
As the sample size increases, the standard deviation of the distribution of sample means increases.