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 5.1.54

Chapter 5, Problem 5.1.54

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z lies between -1.54 and 1.54. This involves using the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

Step 2: Recall that the standard normal distribution table (z-table) provides the cumulative probability P(Z ≤ z) for a given z-value. For this problem, you will need to find the cumulative probabilities for z = 1.54 and z = -1.54.

Step 3: Use the symmetry property of the standard normal distribution. Since the distribution is symmetric about z = 0, the cumulative probability for z = -1.54 is equal to 1 minus the cumulative probability for z = 1.54.

Step 4: Calculate the probability P(-1.54 < z < 1.54) by subtracting the cumulative probability for z = -1.54 from the cumulative probability for z = 1.54. Mathematically, this is expressed as P(-1.54 < z < 1.54) = P(Z ≤ 1.54) - P(Z ≤ -1.54).

Step 5: If using technology (e.g., a calculator or statistical software), input the z-values directly to find the cumulative probabilities and subtract them as described in Step 4. Alternatively, use a z-table to look up the cumulative probabilities and perform the subtraction.

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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 variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it essential for calculating probabilities related to normally distributed data.

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 normal distributions and are crucial for finding probabilities in the standard normal distribution.

Probability Calculation

Probability calculation in the context of the standard normal distribution involves finding the area under the curve between two z-scores. This area represents the probability of a random variable falling within that range. Tools such as z-tables or statistical software can be used to determine these probabilities efficiently, especially for non-standard values.

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Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 150)

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.P(- 1.54 < z < 1.54)

Verified video answer for a similar problem:

Key Concepts

Standard Normal Distribution

Z-scores

Probability Calculation

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)