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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
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.1.51
Chapter 5, Problem 5.1.51

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


P(- 0.89 < z < 0)

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Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z falls between -0.89 and 0. 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 (or z-table) provides the cumulative probability from the far left of the distribution up to a given z-value. Alternatively, technology such as a graphing calculator or statistical software can be used to find probabilities.
Step 3: Break the problem into two parts. First, find the cumulative probability for z = 0. From the properties of the standard normal distribution, the cumulative probability for z = 0 is 0.5 (since it is the mean of the distribution).
Step 4: Next, find the cumulative probability for z = -0.89. Use a z-table or technology to determine this value. The z-table will give the cumulative probability from the far left of the distribution up to z = -0.89.
Step 5: Subtract the cumulative probability for z = -0.89 from the cumulative probability for z = 0. This difference represents the probability that z falls between -0.89 and 0. Mathematically, this is expressed as P(-0.89 < z < 0) = P(z < 0) - P(z < -0.89).

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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 with a mean of 0 and a standard deviation of 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and is often used in statistical analysis to standardize scores.
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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.
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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 likelihood of a value falling within that range. Tools such as z-tables or statistical software can be used to determine these probabilities, making it easier to analyze data and draw conclusions.
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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.


To the left of z= -1.28 and to the right of z= 1.28

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 = 45, sigma =15, n = 100

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 > 65)

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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=20, p=0.65, q=0.35

Textbook Question

Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x≥109)


a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)