Let's break this problem into parts and solve each:

(i) Find the population proportion ($\pi$) of customers that prefer your brand.

In the population, there are 4 customers, and only 1 customer prefers your brand. Therefore, the population proportion $\pi$ is:

$\pi = \frac{\text{Number of customers preferring your brand}}{\text{Total number of customers}} = \frac{1}{4} = 0.25$

So, $\pi = 0.25$.

(ii) Develop the probability distribution of the sample proportion ($p$).

To create the sampling distribution:

1. Population: {Prefers brand ($P$), Does not prefer brand ($D$)} = {1 $P$, 3 $D$}.

2. Sample size: $n = 2$, with replacement.

3. All possible samples:

   • $PP$
   • $PD$
   • $DP$
   • $DD$

   Because sampling is with replacement, $PD$ and $DP$ are considered distinct.

4. Sample Proportion ($p$):

   • For $PP$: Both customers prefer the brand → $p = \frac{2}{2} = 1$.
   • For $PD$ or $DP$: One customer prefers the brand → $p = \frac{1}{2} = 0.5$.
   • For $DD$: No customer prefers the brand → $p = \frac{0}{2} = 0$.

5. Probability distribution:

   • $p = 1$: Only one sample ($PP$) → Probability = $\frac{1}{16}$.
   • $p = 0.5$: Two samples ($PD, DP$) → Probability = (\frac{4*Pd()).

+###Generate plotsd get latter. pd import matplotlib.pyplot as plt

Define the sample proportions and their probabilities

sample_proportions = [0, 0.5, 1] probabilities = [9/16, 6/16, 1/16]

Create a DataFrame for better visualization

data = pd.DataFrame({ 'Sample Proportion (p)': sample_proportions, 'Probability': probabilities })

Display the table of the probability distribution to the user

import ace_tools as tools; tools.display_dataframe_to_user(name="Probability Distribution of Sample Proportions", dataframe=data)

Plot the probability distribution

plt.bar(sample_proportions, probabilities, tick_label=sample_proportions) plt.title("Probability Distribution of Sample Proportion (p)") plt.xlabel("Sample Proportion