6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:21 minutesProblem 5.Q.4a
Textbook Question
The random variable x is normally distributed with the given parameters. Find each probability.
a. μ = 9.2, σ ≈ 1.62, P(x < 5.97)
Verified step by step guidance1
Step 1: Understand the problem. The random variable x is normally distributed with a mean (μ) of 9.2 and a standard deviation (σ) of approximately 1.62. We are tasked with finding the probability P(x < 5.97).
Step 2: Standardize the value of x = 5.97 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (5.97 - 9.2) / 1.62.
Step 3: Simplify the numerator (5.97 - 9.2) and then divide by the standard deviation (1.62) to calculate the z-score. This will give you the standardized value corresponding to x = 5.97.
Step 4: Use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(x < 5.97).
Step 5: Interpret the result. The cumulative probability obtained from the z-table or software is the probability that the random variable x is less than 5.97 in the given normal distribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.
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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. Z-scores are essential for finding probabilities in a normal distribution, as they allow us to convert any normal random variable into a standard normal variable.
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Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes on a value less than or equal to a specific value. For a normally distributed variable, the CDF can be used to find probabilities associated with specific ranges of values, such as P(x < 5.97) in this case. The CDF is crucial for determining probabilities in statistical analysis.
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