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

Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.RE.5

Chapter 5, Problem 5.RE.5

In Exercises 5 and 6, find the area of the indicated region under the standard normal curve. If convenient, use technology to find the area.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the area under the standard normal curve to the left of z = 0.46. This represents the cumulative probability for z-values less than or equal to 0.46.

Step 2: Recall that the standard normal curve is symmetric about z = 0, with a mean of 0 and a standard deviation of 1. The area under the curve represents probabilities.

Step 3: Use the standard normal table (z-table) or technology (such as a graphing calculator or statistical software) to find the cumulative probability corresponding to z = 0.46. The z-table provides the area to the left of a given z-value.

Step 4: If using technology, input the z-value (0.46) into the software or calculator to compute the cumulative probability. Ensure the tool is set to standard normal distribution.

Step 5: Interpret the result. The area under the curve to the left of z = 0.46 represents the probability of a random variable from a standard normal distribution being less than or equal to 0.46.

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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 case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve, and all values are expressed in terms of z-scores, which indicate how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve.

Z-Score

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 allow for the comparison of scores from different distributions and are essential for finding areas under the standard normal curve.

Area Under the Curve

The area under the curve (AUC) in a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be calculated using z-scores and is often found using statistical tables or technology. The AUC is fundamental in hypothesis testing and confidence interval estimation.

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Related Practice

Textbook Question

In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.

Textbook Question

Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

b. You randomly select 15 vials. What is the probability that you select at least one vial that is within the acceptable range?

Textbook Question

Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

b. You randomly select three samples of five vials. What is the probability that you select at least one sample of five vials that has a mean that is within the acceptable range?

Textbook Question

In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(72 < x < 82)

Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = -1.95

Textbook Question

In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(z < 1.28)