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(x < 55)

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 P(x < 55).
2. Step 2: Standardize the value of x = 55 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (55 - 74) / 8.
3. Step 3: Simplify the z-score formula to calculate the z-value. This will give you the standardized value corresponding to x = 55.
4. 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-value. This cumulative probability represents P(x < 55).
5. 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 55 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 for modeling real-world phenomena.

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 in a standard normal distribution table.

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 to convert the values into a standard normal distribution, where probabilities can be easily found using statistical tables or software. For the question at hand, calculating P(x < 55) requires finding the Z-score for x = 55 and then using it to find the corresponding probability.