6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:57 minutesProblem 29
Textbook Question
Designing Helmets Engineers must consider the circumferences of adult heads when designing motorcycle helmets. Adult head circumferences are normally distributed with a mean of 570.0 mm and a standard deviation of 18.3 mm (based on Data Set 3 “ANSUR II 2012â€). Due to financial constraints, the helmets will be designed to fit all adults except those with head circumferences that are in the smallest 5% or largest 5%. Find the minimum and maximum head circumferences that the helmets will fit.
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the minimum and maximum head circumferences that the helmets will fit, excluding the smallest 5% and largest 5% of the distribution. This involves working with a normal distribution and identifying the boundaries corresponding to the middle 90% of the data.
Step 2: Recall the properties of a normal distribution. The mean (μ) is 570.0 mm, and the standard deviation (σ) is 18.3 mm. The smallest 5% and largest 5% correspond to the tails of the distribution, leaving 90% in the middle. To find the boundaries, we need the z-scores that correspond to the 5th percentile and the 95th percentile.
Step 3: Use a z-score table or statistical software to find the z-scores for the 5th percentile and 95th percentile. For a standard normal distribution, the z-score for the 5th percentile is approximately -1.645, and the z-score for the 95th percentile is approximately 1.645.
Step 4: Convert the z-scores to actual head circumferences using the formula for a normal distribution: . For the minimum circumference, use the z-score of -1.645, and for the maximum circumference, use the z-score of 1.645.
Step 5: Substitute the values into the formula. For the minimum circumference: . For the maximum circumference: . Calculate these values to find the minimum and maximum circumferences.
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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 adult head circumferences follow a normal distribution, which allows for the application of statistical methods to determine specific percentiles.
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Percentiles
A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the 5th percentile is the value below which 5% of the data points lie. In the helmet design scenario, identifying the 5th and 95th percentiles of head circumferences helps engineers determine the minimum and maximum sizes for the helmets, ensuring they fit the majority of the adult population.
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. In this case, Z-scores can be used to find the specific head circumference values that correspond to the 5th and 95th percentiles, facilitating the design of helmets that accommodate most adults.
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