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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 12.CR.6a
Chapter 6, Problem 12.CR.6a

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..

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1
Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 5.670 g, and the standard deviation (σ) is 0.062 g. The problem asks for the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g.
Step 2: Standardize the given weights (5.600 g and 5.700 g) into z-scores using the z-score formula: z = (x - μ) / σ. For each weight, substitute the values of x (the weight), μ (the mean), and σ (the standard deviation).
Step 3: Calculate the z-scores for both bounds. For the lower bound (5.600 g), compute z = (5.600 - 5.670) / 0.062. For the upper bound (5.700 g), compute z = (5.700 - 5.670) / 0.062.
Step 4: Use a standard normal distribution table or a statistical software to find the cumulative probabilities corresponding to the calculated z-scores. These cumulative probabilities represent the area under the standard normal curve to the left of each z-score.
Step 5: Subtract the cumulative probability of the lower z-score from the cumulative probability of the upper z-score. This difference gives the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g.

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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, depicting 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 weights of quarters follow a normal distribution, allowing us to use statistical methods to find probabilities related to specific weight ranges.
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Finding Standard Normal Probabilities using z-Table

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 this problem, Z-scores will be used to standardize the weights of the quarters to find the probability of a quarter weighing between 5.600 g and 5.700 g.
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Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. For normally distributed data, this often requires using Z-scores to reference standard normal distribution tables or software to find the area under the curve that corresponds to the desired range of values. In this case, we will calculate the probability that a quarter's weight falls between the specified limits.
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Related Practice
Textbook Question

In Exercises 11–14, use the population of {2, 3, 5, 9} of the lengths of hospital stay (days) of mothers who gave birth, found from Data Set 6 “Births” in Appendix B. Assume that random samples of size n = 2 are selected with replacement.


Sampling Distribution of the Sample Mean


a. After identifying the 16 different possible samples, find the mean of each sample, and then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same. (Hint: See Table 6-3 in Example 2.)

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Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.


Sampling Distribution of the Sample Median


c. Find the mean of the sampling distribution of the sample median.

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Textbook Question

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).

d. If a vending machine is designed to accept quarters with weights above the 10th percentile P10 find the weight separating acceptable quarters from those that are not acceptable.

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Textbook Question

Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.

b. Find the value of Q3, the cell phone radiation amount that is the third quartile.

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Textbook Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):


Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.


Mickey Mouse Disney World requires that people employed as a Mickey Mouse character must have a height between 56 in. and 62 in.


a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as Mickey Mouse characters?

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Textbook Question

Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with μ = 95.5 and σ = 16.0 (based on data from “Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm,” by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected without replacement for a follow-up study.


b. Find the probability that the mean IQ score of the follow-up sample is between 95 and 105.

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