Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 12.CR.6d

Chapter 6, Problem 12.CR.6d

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.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the weight that separates the bottom 10% of quarters (10th percentile, P10) from the rest, assuming the weights are normally distributed with a mean (μ) of 5.670 g and a standard deviation (σ) of 0.062 g.

Step 2: Recall the formula for a z-score in a normal distribution: z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation. We will use this formula to find the weight (X) corresponding to the 10th percentile.

Step 3: Use a z-table or statistical software to find the z-score corresponding to the 10th percentile. The z-score for the 10th percentile (P10) is approximately -1.28. This means that the value of X is 1.28 standard deviations below the mean.

Step 4: Rearrange the z-score formula to solve for X: X = μ + (z * σ). Substitute the known values: μ = 5.670 g, z = -1.28, and σ = 0.062 g.

Step 5: Perform the calculation to find X, which represents the weight separating acceptable quarters from those that are not acceptable. This value will be the 10th percentile weight.

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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, which allows us to use statistical methods to determine percentiles.

Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 10th percentile (P10) is the weight below which 10% of the quarter weights lie. Understanding percentiles is crucial for determining thresholds, such as the weight limit for acceptable quarters in the vending machine.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It allows for the comparison of scores from different distributions. To find the weight corresponding to the 10th percentile, one can calculate the Z-score for P10 and then use it to find the specific weight using the mean and standard deviation of the quarter weights.

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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.)

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.

Snow White Disney World requires that women employed as a Snow White character must have a height between 64 in. and 67 in.

a. Find the percentage of women meeting the height requirement.

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.

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).

a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..

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?

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.