In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.


P(72 < x < 82)

Verified step by step guidance

1. Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 74 and a standard deviation (σ) of 8. The goal is to find the probability that the random variable x falls between 72 and 82, i.e., P(72 < x < 82).
2. Step 2: Standardize the values of x = 72 and x = 82 using the z-score formula: z = (x - μ) / σ. For x = 72, calculate z₁ = (72 - 74) / 8. For x = 82, calculate z₂ = (82 - 74) / 8.
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(72 < x < 82) by subtracting the cumulative probability for z₁ from the cumulative probability for z₂: P(72 < x < 82) = Φ(z₂) - Φ(z₁).
5. Step 5: Interpret the result as the probability that the random variable x falls within the range 72 to 82 in the context of 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). In this distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Understanding this distribution is crucial for calculating probabilities related to normally distributed random variables.

Z-scores

A Z-score represents the number of standard deviations a data point is from the mean of a distribution. It is calculated using the formula Z = (X - mu) / sigma, where X is the value of interest. Z-scores are essential for standardizing different normal distributions, allowing for the comparison of probabilities across different datasets.

Probability Calculation

Calculating probabilities for a normal distribution often involves finding the area under the curve between two points. This is typically done using Z-scores to convert the values into a standard normal distribution, then using statistical tables or software to find the corresponding probabilities. For the given problem, we need to find P(72 < x < 82) by determining the Z-scores for 72 and 82 and calculating the area between them.