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Magazine Chapter 6: Problem 10
Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities. $$ P(40 \leq x \leq 47) ; \mu=50 ; \sigma=15 $$
Short Answer
Expert verified
The probability is approximately 0.1693, or 16.93%.
Step by step solution
01
Understand the Problem
We need to calculate the probability that a normally distributed random variable \(x\) falls between 40 and 47. The distribution has a mean \(\mu = 50\) and a standard deviation \(\sigma = 15\).
02
Standardize the Variable
Convert the variable \(x\) to a standard normal variable (\(z\)). Use the formula: \( z = \frac{x - \mu}{\sigma} \). For \(x = 40\), \( z_1 = \frac{40 - 50}{15} = \frac{-10}{15} = -\frac{2}{3} \). For \(x = 47\), \( z_2 = \frac{47 - 50}{15} = \frac{-3}{15} = -\frac{1}{5} \).
03
Find Probabilities from Standard Normal Distribution
Use the standard normal distribution table (or a calculator) to find \( P(Z < z) \) for each standardized value. For \( z_1 = -\frac{2}{3} \), \( P(Z < -\frac{2}{3}) \approx 0.2514 \). For \( z_2 = -\frac{1}{5} \), \( P(Z < -\frac{1}{5}) \approx 0.4207 \).
04
Calculate the Desired Probability
The probability \( P(40 \leq x \leq 47) \) is found by subtracting the probability at \( z_1 \) from the probability at \( z_2 \): \( P(-\frac{2}{3} < Z < -\frac{1}{5}) = P(Z < -\frac{1}{5}) - P(Z < -\frac{2}{3}) = 0.4207 - 0.2514 = 0.1693 \).
05
Interpret the Results
The probability that \(x\) falls between 40 and 47, given the specified normal distribution, is approximately 0.1693. This means there is a 16.93% chance that a value of \(x\) will be between 40 and 47.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Normal Variable
To understand probability calculations involving normal distributions, we first need to understand the concept of the "standard normal variable." A standard normal variable, denoted by "Z," is a special kind of random variable that follows a normal distribution with a mean of 0 and a standard deviation of 1. This allows us to reference the standard normal distribution, which is universally used to find probabilities and percentiles.
Why do we convert to a standard normal variable? It's quite simple! By transforming our original variable, we can make use of the standard normal distribution table.
Why do we convert to a standard normal variable? It's quite simple! By transforming our original variable, we can make use of the standard normal distribution table.
- This transformation process is called "standardization."
- We use the formula: \( z = \frac{x - \mu}{\sigma} \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.
- Standardization allows us to compare relative positions of different data points within the distribution.
Mean and Standard Deviation
The "mean and standard deviation" are fundamental terms when discussing normal distributions. These two parameters characterize the shape and spread of the distribution.
Let's break them down:
Let's break them down:
- Mean (\(\mu\)): This is the average value of all data points in a distribution. It indicates the central tendency, or where most data points cluster.
- Standard Deviation (\(\sigma\)): This measures the spread or variability of the data around the mean. A larger \(\sigma\) signifies a wider spread, while a smaller \(\sigma\) indicates tighter clustering of data points around the mean.
Probability Calculation
"Probability calculation" in the context of normal distribution involves determining the likelihood of a random variable falling within a specific range. This involves the probability density function (PDF) of the normal distribution.
In this exercise, we're calculating when our random variable \(x\) lies between 40 and 47. Here's how we approach it:
In this exercise, we're calculating when our random variable \(x\) lies between 40 and 47. Here's how we approach it:
- First, standardize \(x\) to a \(z\) value to simplify our calculations. We've already calculated \( z_1 = -\frac{2}{3} \) for \(x = 40\) and \( z_2 = -\frac{1}{5} \) for \(x = 47\).
- Next, we refer to the standard normal distribution table to find probabilities for these \(z\) scores. This provides us with probabilities that \(Z\) is less than a certain value: \( P(Z < -\frac{2}{3}) \approx 0.2514 \) and \( P(Z < -\frac{1}{5}) \approx 0.4207 \).
- Finally, we calculate the probability \( P(40 \leq x \leq 47) \) by subtracting the probabilities: \( P(-\frac{2}{3} < Z < -\frac{1}{5}) = P(Z < -\frac{1}{5}) - P(Z < -\frac{2}{3}) = 0.4207 - 0.2514 = 0.1693 \), which means there is a 16.93% chance that the variable \(x\) will fall within this interval.
