Problem 223 - Applied Statistics and Probability for Engineers

Question

Among homeowners in a metropolitan area, \(25 \%\) recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities: (a) More than 2600 recycle paper in a week (b) Between 2400 and 2600 recycle paper in a week (c) Number of customers who recycle paper in a week that is exceeded with probability approximately 0.05

Step-by-Step Solution

Answer

(a) 0.0104, (b) 0.9792, (c) 2572.

Step 1: Understanding the Distribution

We start by recognizing that this problem can be modeled using a binomial distribution. The number of trials is 10,000 (the number of homeowners), and the probability of success (a homeowner recycling paper) is 0.25.

Step 2: Normal Approximation to the Binomial

Given the large number of trials, we can use the normal approximation to the binomial distribution. Calculate the mean (\( \mu \)) and standard deviation (\( \sigma \)) for the binomial distribution. The mean \( \mu = n \times p = 10000 \times 0.25 = 2500 \). The standard deviation \( \sigma = \sqrt{n \times p \times (1-p)} = \sqrt{10000 \times 0.25 \times 0.75} = \sqrt{1875} \approx 43.3 \).

Step 3: Solving Part (a) - Probability of More than 2600

To find the probability that more than 2600 homeowners recycle, we use the normal approximation and standardize the value. We compute the z-score for 2600 using the formula: \( z = \frac{2600 - 2500}{43.3} \approx 2.31 \). We then look up this z-value in the standard normal distribution table to find the probability. \( P(X > 2600) = 1 - P(Z < 2.31) \approx 1 - 0.9896 = 0.0104 \).

Step 4: Solving Part (b) - Probability Between 2400 and 2600

We calculate the z-scores for both 2400 and 2600. For 2400, \( z = \frac{2400 - 2500}{43.3} \approx -2.31 \). We already calculated the z-score for 2600 as 2.31. Thus, \( P(2400 < X < 2600) = P(-2.31 < Z < 2.31) = P(Z < 2.31) - P(Z < -2.31) \approx 0.9896 - 0.0104 = 0.9792 \).

Step 5: Solving Part (c) - Exceeding Number with Probability 0.05

We need to find \( X \) such that \( P(X > X_0) = 0.05 \) or equivalently \( P(X \leq X_0) = 0.95 \). Using the z-table, a probability of 0.95 corresponds to a z-value of approximately 1.645. We solve \( \frac{X_0 - 2500}{43.3} = 1.645 \), which gives \( X_0 = 1.645 \times 43.3 + 2500 \approx 2571.19 \). Since \( X \) must be an integer, \( X \approx 2572 \).

Key Concepts

Binomial Distribution

The binomial distribution is a key concept in probability theory used to model situations where there are exactly two possible outcomes in each trial, often termed "success" and "failure." Suppose you have multiple independent trials, and each trial has the same probability of success. This is where the binomial distribution shines.
In our example, we have a situation with 10,000 homeowners who either recycle paper (success) or not (failure). Only 25% of the homeowners are expected to recycle, making the probability of success 0.25 for each homeowner. The number of trials here is the total number of homeowners, which is a large number, making this scenario a perfect fit for a binomial setup.
Binomial distributions are defined by two parameters, \( n \) (number of trials) and \( p \) (probability of success). Using these values, computations can be made to determine various probabilities relating to how many successes occur over the given trials. Understanding this concept forms the foundational step before we can apply any approximations or further calculations.

Z-Score Calculation

Z-score calculation is a vital part of using normal approximation to understand and calculate probabilities in distributions. A z-score tells us how many standard deviations an element is from the mean of a distribution. It essentially transforms a random variable from any distribution to the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
In this exercise, to determine the probability of more than 2600 homeowners recycling, we need to compute the z-score using the formula: \( z = \frac{X - \mu}{\sigma} \). Here, \( X \) is the value of interest (e.g., 2600), \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation of the distribution.
Calculating a z-score is crucial because it allows us to use standard normal distribution tables (or software) to find probabilities. These z-values convert our scenario into one that is easier to handle statistically, thereby simplifying the problem significantly.

Probability Approximation

Sometimes, tackling probabilities directly from a binomial distribution can be cumbersome, especially when dealing with a large number of trials. This is where probability approximation, particularly normal approximation, becomes useful.
Since our given scenario involves 10,000 trials, we use a normal approximation to simplify calculations significantly. By approximating our binomial distribution to a normal distribution, we use the calculated mean and standard deviation to find probabilities easily.
Probability approximation lets us make educated guesses about the distribution when exact calculations are challenging due to complexity or size. In this case, for calculating probabilities like finding values between a range (e.g., between 2400 and 2600), we simply calculate the z-scores and refer to the standard normal distribution. This makes the entire probability calculating process efficient and understandable.

Standard Normal Distribution

The standard normal distribution is a key statistical concept that represents a normal distribution with a mean of 0 and a standard deviation of 1. It is depicted as the familiar bell curve and serves as the basis for calculating probabilities through z-scores.
Once we've transformed our values into z-scores, each z-score correspods to a point on this standard curve. By referring to z-tables or software, we can establish the likelihood of an occurrence falling within a certain range.
In our exercise, using the standard normal distribution, we find approximate probabilities, like more than 2600 homeowners recycling, by calculating the z-score (e.g., 2.31) and checking where it falls on the standard curve. This process helps translate specific scenarios like our binomial distribution into a form where probability calculations are more straightforward.