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Questions and Answers
What is the general equation of the normal distribution?
What is the general equation of the normal distribution?
In a normal distribution, what do the parameters μ and σ represent?
In a normal distribution, what do the parameters μ and σ represent?
What characteristic does the normal curve exhibit?
What characteristic does the normal curve exhibit?
What is the total area under the normal curve above the x-axis?
What is the total area under the normal curve above the x-axis?
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If Z = (X - μ) / σ represents a standardized random variable, what does this imply?
If Z = (X - μ) / σ represents a standardized random variable, what does this imply?
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Which statement about the normal distribution is correct?
Which statement about the normal distribution is correct?
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What does the probability density function f(x) describe?
What does the probability density function f(x) describe?
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How do you interpret the standard normal variable Z?
How do you interpret the standard normal variable Z?
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What is the primary function of the distribution function F(z) in the context of normal distribution?
What is the primary function of the distribution function F(z) in the context of normal distribution?
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When calculating probabilities in normal distribution, which of the following statements is NOT true?
When calculating probabilities in normal distribution, which of the following statements is NOT true?
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How would you calculate the probability of a battery cell lasting more than 15 hours given a mean life of 12 hours and a standard deviation of 3 hours?
How would you calculate the probability of a battery cell lasting more than 15 hours given a mean life of 12 hours and a standard deviation of 3 hours?
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In a normal distribution where 31% of items are under 45, what does this percentage represent?
In a normal distribution where 31% of items are under 45, what does this percentage represent?
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If the z-score calculated for x = 64 in a given normal distribution is 1.4, what is the relationship of z to µ and σ?
If the z-score calculated for x = 64 in a given normal distribution is 1.4, what is the relationship of z to µ and σ?
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In the provided examples, what percentage of battery cells is expected to last between 10 and 14 hours?
In the provided examples, what percentage of battery cells is expected to last between 10 and 14 hours?
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Which equation correctly represents the z-score for a given x value in normal distribution?
Which equation correctly represents the z-score for a given x value in normal distribution?
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In the context of normal distribution, what does F(-z₁) equate to according to the given information?
In the context of normal distribution, what does F(-z₁) equate to according to the given information?
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Study Notes
Normal Distribution
- A continuous probability distribution
- Derived from the binomial distribution (large number of trials, probability of success close to 0.5)
- Equation: f(x) = (1 / (σ√(2π))) * e^(-(1/2)((x-µ)/σ)^2)
- x: continuous variable, can take any value from negative infinity to positive infinity
- µ: mean of the distribution
- σ: standard deviation of the distribution
- σ > 0; -∞ < µ < ∞
- Often bell-shaped and symmetrical
- The graph extends to positive and negative infinity along x-axis
- Asymptotic to x-axis (approaches x-axis but never touches it)
- Unimodal (single peak)
- The mean, median, and mode are all the same (µ)
- Area under the curve = 1
- Area represents probability
- Area between two given ordinates represents the probability of values falling into that interval
Standard Form of Normal Distribution
- If X is a normal random variable with mean µ and standard deviation σ, then Z = (X-µ)/σ
- Z has a normal distribution with mean 0 and standard deviation 1 (standard normal variable)
- Probability density function for Z: f(z) = (1 / √(2π)) * e^(-(1/2)z²)
- Useful for finding probabilities relating to normal distributions using standard tables
- Free from parameters of original normal distribution
Illustrative Examples
- Sample of 100 battery cells to determine battery life
- Assume data is normally distributed, calculate percentage with different life spans
- Examples give calculations showing probability (percentage) based on a normal distribution
- Using standard tables to calculate area under the curve.
- Examples on finding mean and standard deviation of normal distributions.
- Illustrating how to translate raw data observations into Z scores, then use a Z-table (standard normal distribution) to determine probabilities.
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Description
Explore the essential concepts of normal distribution, a fundamental continuous probability distribution in statistics. Understand its properties, including the relationship between mean, median, mode, and how to standardize using the Z-score. This quiz will help you grasp the significance of the bell-shaped curve and its applications in probability.
