Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean mu=174 and standard deviation sigma=20. Find the indicated probability.


P(x > 182)

Verified step by step guidance

1. Step 1: Understand the problem. The random variable x follows a normal distribution with mean \( \mu = 174 \) and standard deviation \( \sigma = 20 \). We are tasked with finding the probability \( P(x > 182) \).
2. Step 2: Standardize the value of x = 182 to a z-score using the formula \( z = \frac{x - \mu}{\sigma} \). Substitute \( x = 182 \), \( \mu = 174 \), and \( \sigma = 20 \) into the formula.
3. Step 3: Once the z-score is calculated, use the standard normal distribution table (or a calculator) to find the cumulative probability \( P(Z \leq z) \), where Z is the standard normal variable.
4. Step 4: Since we are looking for \( P(x > 182) \), use the complement rule: \( P(x > 182) = 1 - P(Z \leq z) \). Subtract the cumulative probability from 1.
5. Step 5: Interpret the result. The final value represents the probability that the random variable x is greater than 182 in the given normal distribution.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (mu) and standard deviation (sigma). 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 normal distributions often involves finding the area under the curve for a specified range of values. This is typically done using Z-scores and standard normal distribution tables or software. For the question at hand, calculating P(x > 182) requires determining the Z-score for x = 182 and then finding the corresponding probability that represents the area to the right of this Z-score.