Problem 18 - Basic Statistics for Business and Economics with Student

Question

Can you tell the difference between Coke and Pepsi in a blind taste test? Most people say they can and have a preference for one brand or the other. However, research suggests that people can correctly identify a sample of one of these products only about 60 percent of the time. Suppose we decide to investigate this question and select a sample of 15 college students. a. How many of the 15 students would you expect to correctly identify Coke or Pepsi? b. What is the probability exactly 10 of the students surveyed will correctly identify Coke or Pepsi? C. What is the probability at least 10 of the students will correctly identify Coke or Pepsi?

Step-by-Step Solution

Answer

a. Expect 9 students. b. Probability of exactly 10: 0.215. c. Probability of at least 10: 0.184 + 0.103 + 0.039 + 0.009 + 0.001 + 0.0001 = 0.382.

Step 1: Determine Expected Identifications

The expected number of students who correctly identify Coke or Pepsi is calculated using the success probability and the number of trials. The formula for expectation in a binomial distribution is given by \( E = n \times p \), where \( n = 15 \) (students) and \( p = 0.6 \) (probability of a student correctly identifying the drink). Thus, \( E = 15 \times 0.6 = 9 \).

Step 2: Calculate Probability for Exactly 10 Identifications

To find the probability that exactly 10 students correctly identify the drink, use the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \( n = 15 \), \( k = 10 \), \( p = 0.6 \). Calculate:\[P(X = 10) = \binom{15}{10} (0.6)^{10} (0.4)^5\]Calculate \( \binom{15}{10} \), then multiply by the powers of \( p \) and \( 1-p \).

Step 3: Calculate Probability for At Least 10 Identifications

To calculate the probability that at least 10 students correctly identify the drink, sum the probabilities from 10 to 15 correct identifications: \[P(X \geq 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)\]Use the binomial probability formula for each case (like in Step 2) and add them together.

Key Concepts

Probability Calculation

In probability, calculating the likelihood of certain outcomes is crucial. In the realm of statistics, this often involves using different distributions to model the situation. The binomial distribution is one of these models. It is particularly useful when we have a fixed number of trials, two possible outcomes (like success or failure), and a constant probability of success.
In our exercise, we want to know the probability of exactly 10 students out of 15 correctly identifying Coke or Pepsi. We use the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Where:

• \(n\) is the number of trials (students, here 15)
• \(k\) is the number of successful outcomes (10 students)
• \(p\) is the probability of success on a single trial (0.6 probability of identifying correctly)

Through using this formula, we compute the probability of exactly 10 correct identifications by substituting the values into the formula: \[P(X = 10) = \binom{15}{10} (0.6)^{10} (0.4)^5\] Once you calculate \(\binom{15}{10}\), and the required powers of \(p\) and \(1-p\), you get the probability of this precise event occurring.

Expectation in Probability

Expectation, or the expected value, is a concept in probability that expresses the average outcome you would expect if you repeated the trial many times. For a binomial distribution, the expectation is calculated simply as:
\[E = n \times p\]
Here, \(n\) is the total number of trials, and \(p\) is the probability of success in each trial. In our example with the taste test, we have 15 college students, and a 0.6 probability each one will identify correctly.
Thus, the expected number of students who will identify correctly is \(E = 15 \times 0.6 = 9\).
This result tells you that if you repeated the experiment many times, on average, you would expect 9 out of 15 students to get it right. The expectation gives you insight into the predicted average outcome, even if individual trials can vary.

Sample Size and Probability

Sample size plays a crucial role in statistical experiments involving probabilities. It represents the number of observations or trials performed. In the binomial distribution, each trial is independent, and the outcome is binary (success/failure). Choosing the right sample size is critical because it affects both the accuracy and reliability of the results.
In our exercise, the sample size is 15. With a probability of success of 0.6 (identifying correctly), we calculate expected value and probabilities such as exactly or at least 10 correct identifications. A larger sample size often leads to more reliable results, as variations become more apparent over more trials.
To calculate the probability of at least 10 correct identifications, we sum the probabilities from 10 to the total number of trials (15):
\[P(X \geq 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)\]

• This involves calculating the probability separately for each case using the binomial probability formula and adding them up.

Through this approach, you see the impact of sample size on the overall outcome and ensure comprehensive analysis of all possible favorable scenarios.