The random variable x is normally distributed with the given para...

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

1:39 minutes

Problem 5.Q.4b

Textbook Question

The random variable x is normally distributed with the given parameters. Find each probability.
b. μ = 87, σ ≈ 19, P(x > 40.5)

Verified step by step guidance

Step 1: Understand the problem. The random variable x is normally distributed with a mean (μ) of 87 and a standard deviation (σ) of approximately 19. We are tasked with finding the probability P(x > 40.5).

Step 2: Standardize the value of x = 40.5 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (40.5 - 87) / 19.

Step 3: Simplify the z-score calculation to find the standardized value. This will give you the z-score corresponding to x = 40.5.

Step 4: Use a standard normal distribution table or a statistical software/tool to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(x ≤ 40.5).

Step 5: Since the problem asks for P(x > 40.5), use the complement rule: P(x > 40.5) = 1 - P(x ≤ 40.5). Subtract the cumulative probability from 1 to find the desired probability.

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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, and about 95% falls within two standard deviations. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.

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 allow for the comparison of scores from different normal distributions and are essential for finding probabilities associated with specific values in a normal distribution.

Probability Calculation

Probability calculation in the context of a normal distribution involves determining the likelihood of a random variable falling within a certain range. This is often done using Z-scores and standard normal distribution tables or software. For the given problem, calculating P(x > 40.5) requires finding the Z-score for 40.5 and then using the standard normal distribution to find the corresponding probability.

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The random variable x is normally distributed with the given parameters. Find each probability.b. μ = 87, σ ≈ 19, P(x > 40.5)