Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.1.45

Chapter 5, Problem 5.1.45

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.

Verified step by step guidance

Step 1: Identify the problem. The question asks for the probability of z occurring in the shaded region of the standard normal distribution. From the image, the shaded region lies between z = -1 and z = 1.

Step 2: Understand the standard normal distribution. It is a bell-shaped curve with a mean of 0 and a standard deviation of 1. Probabilities are calculated using the area under the curve.

Step 3: Use the z-scores provided (-1 and 1) to find the cumulative probabilities. The cumulative probability for a z-score represents the area under the curve to the left of that z-score.

Step 4: To find the probability of z occurring in the shaded region, calculate the cumulative probability for z = 1 and subtract the cumulative probability for z = -1. This gives the area between these two z-scores.

Step 5: If convenient, use technology (such as a graphing calculator, statistical software, or online tools) to find the cumulative probabilities for z = -1 and z = 1. Subtract the smaller cumulative probability from the larger one to determine the probability of z occurring in the shaded region.

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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, which is symmetric about the mean. This distribution is crucial for calculating probabilities associated with z-scores, which indicate how many standard deviations an element is from the mean.

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 allow for the comparison of scores from different distributions and are essential for finding probabilities in the standard normal distribution.

Probability Area

The probability area under the curve of the standard normal distribution represents the likelihood of a random variable falling within a specific range of values. In the context of z-scores, the shaded region between two z-scores indicates the probability of a value falling between those scores. This area can be calculated using statistical tables or technology, such as calculators or software.

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Related Practice

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

133

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Textbook Question

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.

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Textbook Question

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

278

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Textbook Question

Explain how to transform a given x-value of a normally distributed variable x into a z-score.

161

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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=24, p=0.85, q=0.15

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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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