91Ó°ÊÓ

Comparing pizza brands \(\quad\) The owners of Aunt Erma's Restaurant plan an advertising campaign with the claim that more people prefer the taste of their pizza (which we'll denote by A) than the current leading fast-food chain selling pizza (which we'll denote by \(\mathrm{D}\) ). To support their claim, they plan to randomly sample three people in Boston. Each person is asked to taste a slice of pizza \(A\) and a slice of pizza \(D\). Subjects are blindfolded so they cannot see the pizza when they taste it, and the order of giving them the two slices is randomized. They are then asked which pizza tastes better. Use a symbol with three letters to represent the responses for each possible sample. For instance, ADD represents a sample in which the first subject sampled preferred pizza \(A\) and the second and third subjects preferred pizza \(\mathrm{D}\) a. Identify the eight possible samples of size \(3,\) and for each sample report the proportion that preferred pizza \(A\). b. In the entire Boston population, suppose that exactly half would prefer pizza \(\mathrm{A}\) and half would prefer pizza D. Explain why the sampling distribution of the sample proportion who prefer Aunt Erma's pizza, when \(n=3,\) is \begin{tabular}{cc} \hline Sample Proportion & Probability \\ \hline 0 & \(1 / 8\) \\ \(1 / 3\) & \(3 / 8\) \\ \(2 / 3\) & \(3 / 8\) \\ 1 & \(1 / 8\) \\ \hline \end{tabular} c. In part b, we can also find the probabilities for each possible sample proportion value using the binomial distribution. Use the binomial with \(n=3\) and \(p=0.50\) to show that the probability of a sample proportion of \(1 / 3\) equals \(3 / 8 .\) (Hint: This equals the probability that \(x=1\) person out of \(n=3\) prefer pizza A. It's especially helpful to use the binomial formula when \(p\) differs from \(0.50,\) since then the eight possible samples listed in part a would not be equally likely.)

Step by step solution

01

Identify Possible Samples

List all possible combinations for three individuals when each can choose either pizza A or pizza D. The combinations are: AAA, AAD, ADA, ADD, DAA, DAD, DDA, DDD.

02

Calculate Proportion for Each Sample

For each combination, calculate the proportion of individuals who preferred pizza A. For example, for AAA, the proportion is 3/3 = 1; for AAD, it's 1/3 = 2/3; etc. The result is as follows: AAA (1), AAD (2/3), ADA (2/3), ADD (1/3), DAA (2/3), DAD (1/3), DDA (1/3), DDD (0).

03

Explain Sampling Distribution

Since each person has a 50% probability of preferring pizza A or D, each of the outcomes in part a has an equal probability of occurring. With n=3, the binomial probability for each configuration must be considered. Thus, the computed proportions and probabilities match the provided table: 0 (1/8), 1/3 (3/8), 2/3 (3/8), 1 (1/8).

04

Use Binomial Distribution

Apply the binomial formula P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n=3, k=1, and p=0.50. For k=1, the probability is P(X=1) = 3C1 * (0.5)^1 * (0.5)^2 = 3 * 0.5 * 0.25 = 3/8, confirming the probability of the 1/3 proportion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sampling Methods

Sampling methods are crucial in statistical analysis as they determine how representative the collected data will be for the entire population. In the exercise, a simple random sample method is used. A simple random sample means each individual within the population has an equal chance of being selected. This ensures that the sample is unbiased and that the results are valid for drawing conclusions about the entire population.
When sampling, it's essential to consider the sample size. In the given scenario, the sample size is three, which is relatively small. However, small samples can still provide useful insights, especially when the sampling process is well-structured and controlled, as demonstrated by the randomization and blinding implemented in the pizza tasting setup.
Randomization in the order of tasting and blinding participants are additional steps that strengthen the randomness and objectivity of the process. By randomizing the tasting order, potential biases due to any tasting sequence are minimized. By blinding the participants, the influence of visual cues on their decisions is eliminated. This approach ensures that the responses are solely based on taste preferences.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent states. In this case, each participant in the sampling process can either prefer pizza A or pizza D. We use the binomial distribution to model this scenario because it deals with the number of successes in a set number of trials, where each trial has only two possible outcomes.
To apply the binomial distribution, we identify the number of trials ( ext{n}=3 ext), which is the sample size, and the probability of success ( ext{p}=0.50 ext), which is the probability that a person will prefer pizza A over pizza D.
The formula for a binomial probability is given by:\[P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\] where \ \(k\) is the number of successes (people preferring pizza A). Using this formula, we can calculate the probability for each possible profile of preferences (0, 1, 2, or 3 people choosing A), ensuring we understand how likely each scenario is within a fair sampling context.

Proportion Calculation

Proportion calculation is a simple yet powerful statistical tool. It helps summarize and represent the diversity within a sample. In this exercise, we're interested in calculating the proportion of participants in each sample who prefer pizza A.
To find the proportion, divide the number of people who choose pizza A by the total sample size. For example, in a sample where two out of three individuals prefer pizza A, the proportion would be \( \frac{2}{3}\). It gives a quantitative measure of the preference for pizza A in that sample.
The calculated proportions from each sample configuration help in constructing the sampling distribution of the sample proportion. This distribution shows the frequency of different possible proportions from our samples, providing insight into how preference for pizza A is distributed among the population. Knowing these proportions from multiple samples allows us to use statistical tools like the binomial distribution to calculate specific probabilities, enhancing our interpretation and conclusions. Understanding proportions and their applications in sampling can greatly enhance decision-making processes and our grasp of population trends.