Estimating Overlapping Preferences for Coffee and Tea

Solution & Explanation

To determine the maximum and minimum possible percentages of the population that enjoy both tea and coffee, we need to analyze the given data using probability theory. Let's break down the information and apply the necessary formulas.

Given Data:

• Sample Size (n): 100 individuals
• P(T): Probability of liking tea = 0.8
• P(C): Probability of liking coffee = 0.7
• P($\neg$T $\cap$ $\neg$C): Probability of liking neither = 0.1

Objective:

Find the range for P(T $\cap$ C), the probability of liking both tea and coffee.

Step-by-Step Calculation:

1. Calculate P(T $\cup$ C):

   • The probability of liking either tea or coffee or both can be calculated as the complement of liking neither.
   • $P(T \cup C) = 1 - P(\neg T \cap \neg C) = 1 - 0.1 = 0.9$

2. Use Inclusion-Exclusion Principle:

   • The formula for the union of two probabilities is: $P(T \cup C) = P(T) + P(C) - P(T \cap C)$
   • Substitute the known values: $0.9 = 0.8 + 0.7 - P(T \cap C)$
   • Simplify to find P(T $\cap$ C): $P(T \cap C) = 0.8 + 0.7 - 0.9 = 0.6$

3. Interpretation:

   • The probability that a randomly selected individual from the sample likes both tea and coffee is 60%.

Confidence Interval Estimation:

To provide a confidence interval for this estimate, we can use the normal approximation to the binomial distribution.

• Standard Error Calculation: $SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.6 \times 0.4}{100}} = 0.049$

• 95% Confidence Interval (Z = 1.96): $CI = p \pm Z \times SE = 0.6 \pm 1.96 \times 0.049$ $CI = (0.6 - 0.096, 0.6 + 0.096) = (0.504, 0.696)$

Conclusion:

• Maximum Possible Percentage: 70% (Assuming everyone who likes coffee also likes tea)
• Minimum Possible Percentage: 0% (Assuming no overlap)
• Estimated Percentage with 95% Confidence Interval: Between 50.4% and 69.6% of the population enjoys both tea and coffee.

These calculations assume that the sample is representative of the broader population and that the survey responses are accurate.