6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:51 minutesProblem 5.CR.16c
Textbook Question
The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months.
c. What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?
Verified step by step guidance1
Identify the problem as finding the value corresponding to the top 5% of a normal distribution. This involves finding the z-score that corresponds to the 95th percentile (since the top 5% is above this point).
Use a z-score table or statistical software to find the z-score corresponding to the 95th percentile. The z-score for the 95th percentile is approximately 1.645.
Apply the z-score formula to relate the z-score to the actual value in the distribution: \( z = \frac{x - \mu}{\sigma} \), where \( z \) is the z-score, \( x \) is the value we are solving for, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
Rearrange the formula to solve for \( x \): \( x = z \cdot \sigma + \mu \). Substitute \( z = 1.645 \), \( \mu = 44 \), and \( \sigma = 5 \) into the formula.
Perform the calculation to find \( x \), which represents the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies.
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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. In this context, the life spans of car batteries follow this distribution, characterized by its bell-shaped curve, where the mean is 44 months and the standard deviation is 5 months.
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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, the Z-score will help determine the life expectancy threshold that corresponds to the top 5% of battery life spans.
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Percentiles
Percentiles are measures that indicate the value below which a given percentage of observations in a group of observations falls. For example, the 95th percentile indicates that 95% of the data points fall below this value. To find the shortest life expectancy in the top 5%, we need to identify the 95th percentile of the normal distribution of battery life spans.
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