Problem 16 - Basic Statistics for Business and Economics with Student

Question

A telemarketer makes six phone calls per hour and is able to make a sale on 30 percent of these contacts. During the next two hours, find: a. The probability of making exactly four sales. b. The probability of making no sales. c. The probability of making exactly two sales. d. The mean number of sales in the two-hour period.

Step-by-Step Solution

Answer

a) P(X = 4) = 0.2311 b) P(X = 0) = 0.0139 c) P(X = 2) = 0.2319 d) Mean = 3.6

Step 1: Identify the Distribution

Since the telemarketer makes six calls per hour and each call has two possible outcomes: making a sale or not making a sale, we can model this situation with a binomial distribution. A Binomial distribution has the form \( \text{Binomial}(n, p) \) where \( n \) is the number of trials and \( p \) is the probability of success in each trial.

Step 2: Define Parameters

The telemarketer makes six calls per hour, so over two hours, \( n = 12 \). Given 30% success rate for each call, the probability of making a sale, \( p = 0.3 \). Hence, the problem is defined by the distribution \( \text{Binomial}(12, 0.3) \).

Step 3: Calculate Probability of Four Sales

For part (a), we want the probability of exactly four sales. Use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \] \[ P(X = 4) = \binom{12}{4} (0.3)^4 (0.7)^8 \] Calculate this value.

Step 4: Calculate Probability of No Sales

For part (b), calculate the probability of making no sales i.e., \( k = 0 \). \[ P(X = 0) = \binom{12}{0} (0.3)^0 (0.7)^{12} \] Simplify and compute the result.

Step 5: Calculate Probability of Two Sales

For part (c), calculate the probability of making exactly two sales, \( k = 2 \). \[ P(X = 2) = \binom{12}{2} (0.3)^2 (0.7)^{10} \] Compute this probability.

Step 6: Calculate the Mean

The mean of a binomial distribution is given by \( \mu = n \times p \). So, the mean number of sales in two hours is \( 12 \times 0.3 = 3.6 \).

Key Concepts

Probability Calculation

Understanding probability helps us determine how likely an event is to occur. In the context of a binomial distribution, we focus on scenarios with two possible outcomes, such as success or failure. For example, a successful sale or no sale.
Using the telemarketer example, we identify three probability calculations:

• Probability of making exactly four sales.
• Probability of making no sales at all.
• Probability of making exactly two sales.

Each is found using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \] where \( n \) is the number of trials, \( k \) is the number of successes, \( p \) is the probability of a success on each trial, and \( 1 - p \) is the probability of a failure. This formula helps calculate any event's probability in a sequence of independent trials.

Mean of a Distribution

The mean, or expected value, of a distribution gives us the average number of successes over many trials. In a binomial distribution, the mean is calculated by multiplying the number of trials by the probability of success.
For the telemarketing scenario, where the telemarketer makes twelve calls, each with a 30% success rate over two hours, the mean number of sales is calculated as follows:\[ \mu = n \times p \] Plugging in the values gives us: \[ \mu = 12 \times 0.3 = 3.6 \] This means that, on average, the telemarketer can expect to make about 3.6 sales in two hours. This theoretical average guides expectations over long periods, though actual results can vary.

Telemarketing Sales

Telemarketing sales are often unpredictable and dynamic. Each call offers only two outcomes: a sale or not. This binary situation aligns well with binomial distribution modeling, offering a structured approach to understanding and predicting sales performance.
In practical terms, understanding these probabilities can help telemarketers set realistic targets. It can guide training and improve strategies. By calculating the mean of sales, telemarketers can evaluate performance against what is statistically predicted, allowing for adjustments.

• Helps in setting achievable goals.
• Provides performance benchmarks.
• Enables strategic planning for future calls.

Statistical tools transform seemingly random outcomes into actionable insights, vital for maximizing efficiency and efficiency in telemarketing.