Question

Suppose that \(50 \%\) of all young adults prefer McDonald's to Burger King when asked to state a preference. A group of 10 young adults were randomly selected and their preferences recorded. a. What is the probability that more than 6 preferred McDonald's? b. What is the probability that between 4 and 6 (inclusive) preferred McDonald's? c. What is the probability that between 4 and 6 (inclusive) preferred Burger King?

Short answer

Question: What are the probabilities for each of the following scenarios: a) More than 6 young adults preferring McDonald's b) Between 4 and 6 young adults (inclusive) preferring McDonald's c) Between 4 and 6 young adults (inclusive) preferring Burger King Answer: To find the probabilities for the given scenarios, use the binomial probability formula for each possible value of k, and then sum up the probabilities: a) P(X>6) = P(X=7) + P(X=8) + P(X=9) + P(X=10) b) P(4 ≤ X ≤ 6) = P(X=4) + P(X=5) + P(X=6) c) Since the probability of preferring McDonald's is 50%, the probability distribution is the same for both McDonald's and Burger King. Thus, the answer to part c will be the same as the answer to part b: P(4 ≤ X ≤ 6) = P(X=4) + P(X=5) + P(X=6).

Step-by-step solution

Calculate probabilities for 7, 8, 9, and 10 young adults preferring McDonald's

We will calculate the probabilities for 7, 8, 9, and 10 young adults preferring McDonald's individually and then sum these probabilities: $$ P(X>6) = P(X=7) + P(X=8) + P(X=9) + P(X=10) $$

Calculate probability for each k (7, 8, 9, 10)

Use the binomial probability formula to calculate the probability for each k: $$ P(X=k) = C(n,k) \cdot p^k \cdot (1-p)^{n-k} $$ Where n = 10, p = 0.50, and k takes the values 7, 8, 9, and 10. Then sum up the probabilities. b. Probability that between 4 and 6 preferred McDonald's

Calculate probabilities for 4, 5, and 6 young adults preferring McDonald's

We will calculate the probabilities for 4, 5, and 6 young adults preferring McDonald's individually and then sum these probabilities: $$ P(4 \leq X \leq 6) = P(X=4) + P(X=5) + P(X=6) $$

Calculate probability for each k (4, 5, 6)

Use the binomial probability formula to calculate the probability for each k: $$ P(X=k) = C(n,k) \cdot p^k \cdot (1-p)^{n-k} $$ Where n = 10, p = 0.50, and k takes the values 4, 5, and 6. Then sum up the probabilities. c. Probability that between 4 and 6 preferred Burger King

Realize that probability of 4 to 6 young adults preferring Burger King is the same as the probability of 4 to 6 young adults preferring McDonald's

Since the probability of preferring McDonald's is 50% and the probability of preferring Burger King is also 50%, the probability distribution will be the same for both. Thus, the answer to part c will be the same as the answer to part b. In conclusion, probabilities for each part are calculated by summing up the probabilities for each possible value of k based on the binomial probability formula.

Key concepts

Probability Calculation

The concept of probability calculation is fundamental in determining the likelihood of an event happening. In our scenario with McDonald's and Burger King, you are determining the probability of specific numbers of young adults selecting one over the other. Probability calculation requires a clear understanding of the total possible outcomes and the favorable outcomes.
For our exercise, since we are interested in young adults preferring McDonald's, each person's choice is considered an independent event. This means the preference of one individual does not affect the others. The calculation process becomes methodical—where each probability for a potential outcome (like preferring McDonald's) is calculated and summed if needed. Such calculations usually involve binomial probability formulas, which we'll discuss further later.

Random Selection

Random selection ensures that each individual in a population has an equal chance of being chosen. This helps eliminate bias and creates a sample that closely represents the entire population's characteristics.
In our problem, selecting 10 young adults at random means any young adult has the potential to be selected, making it a non-biased representative sampling of preferences. The use of random selection is crucial as it gives authenticity to the data collected and leads to more reliable statistical conclusions. Additionally, the randomness forms the basis for applying the binomial distribution, which assumes fixed trials with only two possible outcomes.

Binomial Probability Formula

The binomial probability formula helps us determine the probability of a specific number of successes in a series of independent trials. For our exercise, this formula is crucial. Each young adult’s restaurant preference is a single trial with two possible outcomes: McDonald's or Burger King.
The formula is:\[P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \]where \(n\) is the total number of trials, \(k\) represents the number of successes (young adults preferring McDonald's), and \(p\) is the probability of one success (one preference for McDonald's). Utilizing this formula, you compute the probability for each possible number of successes (4, 5, 6, etc.) and if needed, sum these probabilities to find totals for groups of outcomes, such as "more than 6" or "between 4 to 6."

Statistical Analysis

Statistical analysis involves interpreting data, in this case, the preference of young adults between two fast food chains. Analysis uses calculated probabilities to derive conclusions from your data. In our example, once you've calculated the probabilities using the binomial formula, you perform statistical analysis by comparing the results for different groups (such as 4 to 6 McDonald's preferences, or more than 6). This helps clarify the likely distribution of preferences among the group. Statistical analysis can reveal patterns and offer insights into the sample data, which can potentially be extrapolated to larger populations, understanding the general preference trends among young adults. Such analytical methods offer invaluable assistance in decision-making and strategic planning.