6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:57 minutesProblem 6.CR.3a
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.
Verified step by step guidance1
Step 1: Identify the given parameters of the normal distribution. The mean (μ) is 246.3 mm, and the standard deviation (σ) is 12.4 mm. The problem asks for the probability that a randomly selected adult female has a foot length less than 221.5 mm.
Step 2: Standardize the value 221.5 mm to a z-score using the z-score formula: z = (X - μ) / σ. Here, X is the value of interest (221.5 mm), μ is the mean (246.3 mm), and σ is the standard deviation (12.4 mm). Substitute the values into the formula.
Step 3: Once the z-score is calculated, use a standard normal distribution table (z-table) or a statistical software to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the probability that a randomly selected adult female has a foot length less than 221.5 mm.
Step 4: Interpret the cumulative probability obtained from the z-table or software. This value is the probability that a randomly selected adult female has a foot length less than 221.5 mm.
Step 5: If required, express the probability as a percentage by multiplying the cumulative probability by 100. This step is optional and depends on how the final answer needs to be presented.
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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 foot lengths of adult females follow a normal distribution, which allows us to use statistical methods to find probabilities related to specific values.
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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 question, calculating the Z-score for a foot length of 221.5 mm will help determine how many standard deviations this value is from the mean, which is essential for finding the corresponding probability in the normal distribution.
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Probability
Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In the context of this question, we are interested in finding the probability that a randomly selected adult female has a foot length less than 221.5 mm. This involves using the Z-score to reference standard normal distribution tables or software to find the cumulative probability associated with that Z-score.
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