6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:09 minutesProblem 6.R.8a
Textbook Question
In Exercises 8 and 9, assume that women have standing eye heights that are normally distributed with a mean of 59.7 in. and a standard deviation of 2.5 in. (based on anthropometric survey data from Gordon, Churchill, et al.).
a. If an eye recognition security system is positioned at a height that is uncomfortable for women with standing eye heights less than 54 in., what percentage of women will find that height uncomfortable?
Verified step by step guidance1
Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 59.7 inches, and the standard deviation (σ) is 2.5 inches. The problem asks for the percentage of women with standing eye heights less than 54 inches.
Step 2: Standardize the value of 54 inches using the z-score formula: z = (X - μ) / σ. Here, X is the value of interest (54 inches), μ is the mean (59.7 inches), and σ is the standard deviation (2.5 inches). Substitute these values into the formula.
Step 3: Once the z-score is calculated, use a standard normal distribution table (z-table) or statistical software to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the proportion of women with standing eye heights less than 54 inches.
Step 4: Convert the cumulative probability into a percentage by multiplying it by 100. This percentage represents the proportion of women who will find the eye recognition system height uncomfortable.
Step 5: Interpret the result in the context of the problem. State that this percentage represents the proportion of women whose standing eye heights are less than 54 inches, making the system height uncomfortable for them.
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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, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this context, women's standing eye heights are normally distributed, which allows us to use statistical methods to determine probabilities related to specific height thresholds.
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Z-Score
A Z-score measures how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this problem, calculating the Z-score for the height of 54 inches will help determine the percentage of women whose eye heights fall below this threshold.
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Percentile and Area Under the Curve
The percentile indicates the relative standing of a value within a dataset, showing the percentage of observations that fall below it. In the context of a normal distribution, the area under the curve to the left of a specific Z-score represents the percentage of women with eye heights less than that value, which is essential for answering the question about discomfort with the security system's height.
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