In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6

Find each probability.

b. P(0 < x < 5)

검증된 단계별 안내

Step 1: Understand the problem. The random variable x is normally distributed with a mean (μ) of 18 and a standard deviation (σ) of 7.6. We are tasked with finding the probability that x lies between 0 and 5, i.e., P(0 < x < 5).

Step 2: Standardize the values of x = 0 and x = 5 using the z-score formula: z = (x - μ) / σ. For x = 0, calculate z₀ = (0 - 18) / 7.6. For x = 5, calculate z₅ = (5 - 18) / 7.6.

Step 3: Use the standard normal distribution table (or a calculator) to find the cumulative probabilities corresponding to z₀ and z₅. These probabilities represent the areas under the standard normal curve to the left of z₀ and z₅, respectively.

Step 4: To find P(0 < x < 5), subtract the cumulative probability for z₀ from the cumulative probability for z₅. Mathematically, P(0 < x < 5) = P(z₅) - P(z₀).

Step 5: Interpret the result. The value obtained represents the probability that the random variable x falls between 0 and 5 in the given normal distribution.

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주요 개념

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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, and about 95% falls within two standard deviations. 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, mu is the mean, and sigma is the standard deviation. Z-scores are essential for finding probabilities in a normal distribution, as they allow us to use standard normal distribution tables or software to determine the likelihood of a given range of values.

Probability Calculation

Probability calculation in statistics involves determining the likelihood of a specific event occurring within a defined range. For normally distributed variables, this often requires converting the values into Z-scores and then using the standard normal distribution to find the corresponding probabilities. In the context of the question, calculating P(0 < x < 5) involves finding the probabilities associated with the Z-scores for 0 and 5 and then determining the area under the curve between these two points.

In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6Find each probability.b. P(0 < x < 5)