Problem 45 - Basic Statistics for Business & Economics

Question

Unilever Inc. recently developed a new body wash with a scent of ginger. Their research indicates that \(30 \%\) of men like the new scent. To further investigate, Unilever's marketing research group randomly selected 15 men and asked them if they liked the scent. What is the probability that six or more men like the ginger scent in the body wash?

Step-by-Step Solution

Answer

The probability that six or more men like the ginger scent is approximately 0.2311.

Step 1: Identify the Probability Distribution

This problem involves a probability distribution. Since each man has a binary outcome (like or dislike the scent), this scenario is suitable for a binomial distribution. Key parameters include: number of trials (n=15) and probability of success (p=0.3). The probability we want to find is that 6 or more men like the scent.

Step 2: Express the Probability Using the Binomial Formula

The probability of exactly k successes in a binomial distribution is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). We are interested in finding \( P(X \geq 6) \), which can be expressed as the sum of probabilities from 6 to 15: \( P(X \geq 6) = P(X=6) + P(X=7) + \cdots + P(X=15) \).

Step 3: Calculate Individual Probabilities

Calculate \( P(X = k) \) for each k from 6 to 15 using \( P(X = k) = \binom{15}{k} (0.3)^k (0.7)^{15-k} \). This involves repeated calculations for each k value. Alternatively, use technology or statistical tables to compute this more efficiently.

Step 4: Sum the Probabilities

Sum the probabilities that you calculated previously from \( P(X=6) \) to \( P(X=15) \). This will yield the cumulative probability \( P(X \geq 6) \).

Step 5: Final Probability Calculation

Using a statistical calculator or software, you can directly compute \( P(X \geq 6) \) to find the probability that 6 or more men like the ginger scent. The calculation gives approximately 0.2311.

Key Concepts

Probability Calculation

In the binomial distribution context, probability calculation is about determining the likelihood of achieving a certain number of successes across a series of independent trials. Here, each trial refers to asking one man whether he likes the ginger scent.
You can calculate the probability of "exact" outcomes using the binomial formula: \[ \ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:

• \( n \) is the total number of trials (15 men)
• \( k \) is the number of "successes" you are interested in
• \( p \) is the probability of success for each trial (0.3 in this case)

To find the probability that 6 or more men like the scent, calculate the probabilities for 6, 7, and so on up to 15, and sum them.

Cumulative Probability

Cumulative probability is the sum of individual probabilities for a range of outcomes. In our case, we're interested in the cumulative probability of 6 or more men liking the ginger scent.
This means adding up the individual probabilities for each outcome from 6 through 15: \[ P(X \geq 6) = P(X=6) + P(X=7) + \ldots + P(X=15) \] Calculating each term separately can be tedious, but it accurately gives the total probability of these collective events happening. In practical scenarios, statistical software or calculators can perform this cumulative probability calculation efficiently.

Binary Outcome

Every question asked in this survey results in a binary outcome: either the man likes the scent or he does not. This is what makes the situation ideal for a binomial distribution, where each trial can only result in two possible outcomes.
Consider the binary outcomes in this exercise:

• Success: the man likes the scent (probability \( p = 0.3 \))
• Failure: the man does not like the scent (probability \( 1-p = 0.7 \))

This binary nature allows us to apply the binomial distribution formula to assess the variability and frequency of success across multiple independent trials.

Statistical Tables

Statistical tables are tools often used to simplify probability calculations, especially in binomial distributions. These tables can provide the probability of a given number of successes without computing each individual outcome by hand.
In the context of this exercise, a binomial table that lists the cumulative probabilities for \( n = 15 \) and \( p = 0.3 \) can quickly show \( P(X \geq 6) \).
However, with modern technology, many prefer online calculators or software like Excel that can handle these calculations faster and more accurately, avoiding manual errors and providing results at the click of a button. These technological aids are crucial for swiftly dealing with larger datasets or more complex probability distributions.