The random variable x is normally distributed with the given parameters. Find each probability.

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

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Step 1: Understand the problem. The random variable x follows a normal distribution with mean (μ) = 5.5 and standard deviation (σ) ≈ 0.08. We are tasked with finding the probability that x lies between 5.36 and 5.64, i.e., P(5.36 < x < 5.64).

Step 2: Standardize the values of x to convert them into z-scores using the formula: z = (x - μ) / σ. For the lower bound (x = 5.36), calculate z₁ = (5.36 - 5.5) / 0.08. For the upper bound (x = 5.64), calculate z₂ = (5.64 - 5.5) / 0.08.

Step 3: Use the standard normal distribution table (or a calculator) to find the cumulative probabilities corresponding to z₁ and z₂. Let Φ(z) represent the cumulative probability for a given z-score. Find Φ(z₁) and Φ(z₂).

Step 4: Compute the probability P(5.36 < x < 5.64) by subtracting the cumulative probability at z₁ from the cumulative probability at z₂. This can be expressed as: P(5.36 < x < 5.64) = Φ(z₂) - Φ(z₁).

Step 5: Interpret the result. The value obtained represents the probability that the random variable x falls within the range 5.36 to 5.64 under 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 (μ) and standard deviation (σ). 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.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. To find probabilities for any normal distribution, we often convert the values to the standard normal distribution using the z-score formula: z = (x - μ) / σ. This transformation allows us to use standard normal distribution tables or software to find probabilities associated with specific ranges of values.

Probability Calculation

Calculating probabilities for a normal distribution involves finding the area under the curve between two points. For the given parameters, P(5.36 < x < 5.64) can be determined by calculating the z-scores for both values and then using the standard normal distribution to find the corresponding probabilities. The difference between these probabilities gives the desired probability for the range specified.

The random variable x is normally distributed with the given parameters. Find each probability.c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)