Problem 45 - Basic Statistics for Business and Economics with Student

Question

Recent statistics suggest that 15 percent of those who visit Blair, Inc., an online retail clothier, make a purchase. Blair's planning department wishes to verify this claim. To do so, they selected a sample of 16 "hits" to the site and found that 4 actually made a purchase. a. What is the likelihood of exactly four purchases? b. How many purchases should they expect? c. What is the likelihood that four or more "hits" result in a purchase?

Step-by-Step Solution

Answer

a. Likelihood of exactly 4 purchases is 0.1913. b. Expected purchases are 2.4. c. Likelihood of 4 or more purchases is 0.2603.

Step 1: Identify the distribution

The scenario involves a fixed number of trials (16 hits), where each trial represents an independent event of a customer making a purchase with a probability of 15% or 0.15. This setup is perfect for a binomial distribution.

Step 2a: Calculate likelihood of exactly four purchases

Use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here, \(n = 16\) (total trials), \(k = 4\), and \(p = 0.15\). Calculate \( \binom{16}{4} \cdot 0.15^4 \cdot 0.85^{12} \).

Step 3a: Calculate binomial coefficient

Calculate \( \binom{16}{4} \) which equals 1820. This represents the number of ways to choose 4 purchases from 16 trials.

Step 4a: Complete likelihood calculation for 4 purchases

Calculate the probability: \[ P(X = 4) = 1820 \cdot (0.15)^4 \cdot (0.85)^{12} \]This results in \( P(X = 4) \approx 0.1913 \).

Step 2b: Calculate expected number of purchases

The expected number of purchases in a binomial distribution is given by the formula: \[ E(X) = n \cdot p \].For this case, it is \(16 \times 0.15 = 2.4\).

Step 2c: Calculate likelihood of 4 or more purchases

This involves calculating \(P(X \geq 4)\). It can be computed by summing up \( P(X=4), P(X=5), \ldots, P(X=16) \) or using the complement rule: \[ P(X \geq 4) = 1 - P(X < 4) \].Calculate \(P(X < 4)\) and subtract from 1.

Step 3c: Calculate probabilities for 0, 1, 2, and 3 purchases

Calculate each probability using the binomial formula for \(k = 0, 1, 2, 3\). Sum these to find \(P(X < 4)\).

Step 4c: Complete computation of likelihood for 4 or more purchases

Calculate \(P(X < 4)\) which equals approximately 0.7397. Thus, \(P(X \geq 4) = 1 - 0.7397 = 0.2603\).

Key Concepts

Probability Calculation

Understanding probability calculation is crucial when dealing with a binomial distribution. In this context, probability calculation refers to determining the likelihood of a certain number of successes in a series of trials. The formula for binomial probability is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where:

• \( n \) is the total number of trials.


• \( k \) is the number of successful outcomes you're interested in.


• \( p \) is the probability of success in a single trial.


• \( (1-p) \) is the probability of failure.

For example, in the provided exercise, we calculated the probability of exactly four purchases out of 16 hits with a 15% success rate per hit. The binomial coefficient, also written as \( \binom{n}{k} \), represents the number of ways to choose \( k \) successes out of \( n \) trials. This calculation provides us clarity on how likely we are to observe specific numbers of successful purchase events given our trials and probability.

Expected Value

The expected value in a binomial distribution provides a measure of the average outcome over many trials. It's like having a long-term view of your data, predicting the number of times the event will occur if repeated multiple times. To find the expected value, we use the formula:\[ E(X) = n \cdot p \]Where:

• \( n \) is the total number of trials.


• \( p \) is the probability of success for each trial.

In the mentioned exercise, with 16 trials and a success probability of 0.15, the expected number of purchases was calculated to be 2.4. This implies, on average, that Blair, Inc. should expect around 2 to 3 purchases per 16 site visitors. Remember, the expected value is a theoretical prediction, not a guarantee of exact results in every instance. However, it provides a useful benchmark.

Probability Distribution

Probability distribution describes how the probabilities of different possible outcomes are distributed for a random variable. In the case of a binomial distribution, it summarizes the likelihood of obtaining each possible number of successes.For a binomial distribution, each outcome's probability is based on the binomial formula. In the original exercise, each possible number of purchases (from 0 to 16) has an associated probability, creating a distribution. We calculated the probability for fewer than 4 purchases to find the likelihood of at least 4 or more purchases, making use of the complement rule: \[ P(X \geq 4) = 1 - P(X < 4) \] The analysis showed that while more than half the sample resulted in a purchase, less than 30% of these scenarios resulted in 4 or more purchases. This probability distribution provides a comprehensive view of all possible outcomes in the sample.