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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 6.CR.3c
Chapter 6, Problem 6.CR.3c

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).


c. Find P95.

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1
Step 1: Understand the problem. The question asks for P95, which represents the 95th percentile of a normal distribution. This means we need to find the value of foot length (x) such that 95% of the data lies below it. The distribution is normal with a mean (μ) of 246.3 mm and a standard deviation (σ) of 12.4 mm.
Step 2: Recall the formula for converting a raw score (x) to a z-score in a normal distribution: z = (x - μ) / σ. To find P95, we will reverse this formula to solve for x: x = z * σ + μ.
Step 3: Look up the z-score corresponding to the 95th percentile in a standard normal distribution table or use statistical software. The z-score for the 95th percentile is approximately 1.645.
Step 4: Substitute the known values into the formula x = z * σ + μ. Here, z = 1.645, μ = 246.3 mm, and σ = 12.4 mm. The formula becomes x = (1.645 * 12.4) + 246.3.
Step 5: Simplify the expression to calculate the value of x, which represents P95. This will give the foot length at the 95th percentile.

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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, foot lengths of women are assumed to follow a normal distribution, which allows for the application of statistical methods to analyze the data.
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Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, P95 (the 95th percentile) represents the foot length below which 95% of the adult female population's foot lengths fall. Understanding percentiles is crucial for interpreting data distributions and making comparisons within a dataset.

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 is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for finding percentiles in a normal distribution, as they allow us to determine how far a specific value is from the mean and to use standard normal distribution tables.
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Related Practice
Textbook Question

Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.


About __ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).

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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30


h. Are the wait times discrete data or continuous data?

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Textbook Question

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).


a. Find the probability that a randomly selected adult female has a foot length less than 221.5 mm.

149
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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.

a. Find the probability that a randomly selected cell phone has a radiation amount that exceeds the U.S. standard of 1.6 W/kg or less.

115
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Textbook Question

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).


d. Find the probability that 16 adult females have foot lengths with a mean greater than 250 mm.

108
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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30 


b. Find the median.

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