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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.RS.1b
Chapter 5, Problem 5.RS.1b

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?

Verified step by step guidance
1
Step 1: Understand the problem. The machine has shifted, and the new distribution of the compound added to vials has a mean of 9.96 milligrams and a standard deviation of 0.05 milligrams. You are tasked with finding the probability of selecting at least one vial within the acceptable range when randomly selecting 15 vials.
Step 2: Define the acceptable range. From the graph, the upper limit of the acceptable range is clearly marked. You need to determine the z-score for the upper limit using the formula: z=x-μσ, where x is the upper limit, μ is the mean, and σ is the standard deviation.
Step 3: Calculate the probability for one vial being within the acceptable range. Use the z-score obtained in Step 2 to find the corresponding cumulative probability from the standard normal distribution table. This gives the probability of a single vial being within the acceptable range.
Step 4: Use the complement rule to find the probability of a vial being outside the acceptable range. Subtract the probability of being within the range from 1. This gives the probability of a single vial being outside the range.
Step 5: Calculate the probability of at least one vial being within the acceptable range when selecting 15 vials. Use the formula for the complement of the binomial probability: P(at least 1 vial within range)=1-(probability of being outside)15. This formula accounts for the probability of all 15 vials being outside the range.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the distribution of the compound amounts in vials follows a normal distribution, which is crucial for calculating probabilities related to the selected vials.
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Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this scenario, the standard deviation of 0.05 milligrams indicates how much the amounts of the compound in the vials vary from the mean of 9.96 milligrams.
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Probability of Events

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this question, we are interested in the probability of selecting at least one vial within an acceptable range from a sample of 15 vials. This involves calculating the complement of the probability of selecting no vials within that range, which can be derived from the normal distribution properties and the z-scores corresponding to the acceptable limits.
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Related Practice
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.


c. Which is more sensitive to a shift of parameters—an individual random selection or a randomly selected sample mean?

Textbook Question

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


Textbook Question

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.


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.



a. What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)

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 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