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 5.1.42

Chapter 5, Problem 5.1.42

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: Understand the problem. The goal is to find the probability of z occurring in the shaded region of the standard normal distribution. The shaded region is between z = 0 and z = 1.96.

Step 2: Recall that the standard normal distribution has a mean of 0 and a standard deviation of 1. The z-scores represent the number of standard deviations away from the mean.

Step 3: Use the cumulative probability function for the standard normal distribution to find the area under the curve from z = 0 to z = 1.96. This can be done using statistical tables (z-tables) or technology such as a calculator or software.

Step 4: Look up the cumulative probability for z = 1.96 in the z-table or calculate it using technology. This value represents the area under the curve from z = -∞ to z = 1.96.

Step 5: Subtract the cumulative probability for z = 0 (which is 0.5, as it is the midpoint of the standard normal distribution) from the cumulative probability for z = 1.96 to find 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 z-score, which indicates how many standard deviations an element is from the mean. The area under the curve represents probabilities, with the total area equaling 1.

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. In the context of the standard normal distribution, z-scores allow us to determine the probability of a value falling within a certain range.

Probability and Area Under the Curve

In statistics, the probability of a z-score occurring within a certain range in the standard normal distribution is represented by the area under the curve for that range. This area can be calculated using z-tables or technology, and it reflects the likelihood of a random variable falling within that specified range.

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Related Practice

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(55 < x < 60)

191

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Textbook Question

Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.

Find the probability that the claim will be rejected, assuming that the manufacturer’s claim is true.

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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)

156

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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 right of z= -0.355

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Textbook Question

Milk Containers A machine is set to fill milk containers with a mean of 64 ounces and a standard deviation of 0.11 ounce. A random sample of 40 containers has a mean of 64.05 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain.

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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 ≥ 110)

112

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