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(172 < x <192)

Verified step by step guidance

1. Step 1: Understand the problem. The random variable x follows a normal distribution with mean μ = 174 and standard deviation σ = 20. We are tasked with finding the probability that x lies between 172 and 192, i.e., P(172 < x < 192).
2. Step 2: Standardize the values of x = 172 and x = 192 using the z-score formula: z = (x - μ) / σ. For x = 172, calculate z₁ = (172 - 174) / 20. For x = 192, calculate z₂ = (192 - 174) / 20.
3. Step 3: Use the z-scores obtained in Step 2 to find the corresponding cumulative probabilities from the standard normal distribution table or a statistical software. Let Φ(z₁) represent the cumulative probability for z₁ and Φ(z₂) represent the cumulative probability for z₂.
4. Step 4: Compute the probability P(172 < x < 192) by subtracting the cumulative probability for z₁ from the cumulative probability for z₂: P(172 < x < 192) = Φ(z₂) - Φ(z₁).
5. Step 5: Interpret the result. The value obtained in Step 4 represents the probability that the random variable x falls between 172 and 192 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-scores

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 involves determining the likelihood of a random variable falling within a specific range. This is typically done using Z-scores to convert the values into standard normal variables, which can then be referenced against standard normal distribution tables or calculated using statistical software. For the given range P(172 < x < 192), the probabilities for the corresponding Z-scores must be found and subtracted to obtain the final probability.