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)

Verified step by step guidance

1. Step 1: Recognize that the problem involves a normal distribution with mean (μ) = 5.5 and standard deviation (σ) ≈ 0.08. The goal is to find the probability that the random variable x lies between 5.36 and 5.64, i.e., P(5.36 < x < 5.64).
2. Step 2: Standardize the values 5.36 and 5.64 using the z-score formula: z = (x - μ) / σ. For x = 5.36, calculate z₁ = (5.36 - 5.5) / 0.08. For x = 5.64, calculate z₂ = (5.64 - 5.5) / 0.08.
3. Step 3: Use a standard normal distribution table or a statistical software to find the cumulative probabilities corresponding to z₁ and z₂. 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(5.36 < x < 5.64) by subtracting the cumulative probability for z₁ from the cumulative probability for z₂: P(5.36 < x < 5.64) = Φ(z₂) - Φ(z₁).
5. Step 5: Interpret the result as the probability that the random variable x falls within the specified range, based on the standard normal distribution.

Key Concepts

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 can convert the values to z-scores using the formula z = (x - μ) / σ. This transformation allows us to use standard normal distribution tables or software to find probabilities associated with any normal distribution.

Probability Calculation

To calculate the probability of a random variable falling within a specific range in a normal distribution, we find the z-scores for the endpoints of the range and then use the standard normal distribution to determine the area under the curve between these z-scores. This area represents the probability that the random variable falls within the specified range, which can be found using cumulative distribution functions or z-tables.