Questions: Which value of p̂ appears to give the maximum value of the product p̂q? Choose the correct answer below. A. The value p̂=0.49 gives the maximum value of the product p̂q̂. B. The value p̂=0.55 gives the maximum value of the product p̂q̂. C. The value p̂=0.51 gives the maximum value of the product p̂q̂. D. The value p̂=0.50 gives the maximum value of the product p̂q̂.

Which value of p̂ appears to give the maximum value of the product p̂q? Choose the correct answer below.
A. The value p̂=0.49 gives the maximum value of the product p̂q̂.
B. The value p̂=0.55 gives the maximum value of the product p̂q̂.
C. The value p̂=0.51 gives the maximum value of the product p̂q̂.
D. The value p̂=0.50 gives the maximum value of the product p̂q̂.

Transcript text: Which value of $\hat{p}$ appears to give the maximum value of the product $\hat{p q} ?$ Choose the correct answer below. A. The value $\hat{p}=0.49$ gives the maximum value of the product $\hat{p} \hat{q}$. B. The value $\hat{p}=0.55$ gives the maximum value of the product $\hat{p} \hat{q}$. C. The value $\hat{p}=0.51$ gives the maximum value of the product $\hat{p} \hat{q}$. D. The value $\hat{p}=0.50$ gives the maximum value of the product $\hat{p} \hat{q}$.

Solution

Solution Steps

To find the value of $\hat{p}$ that maximizes the product $\hat{p} \hat{q}$, where $\hat{q} = 1 - \hat{p}$, we need to express the product as a function of $\hat{p}$ and find its maximum. The product $\hat{p} \hat{q}$ can be written as $\hat{p} (1 - \hat{p})$. This is a quadratic function, and its maximum value occurs at the vertex. For a quadratic function $ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$. Here, $a = -1$ and $b = 1$, so the maximum occurs at $\hat{p} = 0.5$.

Step 1: Define the Product Function

We start by defining the product function $ P(\hat{p}) = \hat{p} \cdot \hat{q} $, where $ \hat{q} = 1 - \hat{p} $. Thus, we can express the product as: \[ P(\hat{p}) = \hat{p} (1 - \hat{p}) = \hat{p} - \hat{p}^2 \]

Step 2: Evaluate the Product at Given Values

Next, we evaluate the product $ P(\hat{p}) $ at the specified values of $ \hat{p} $:

For $ \hat{p} = 0.49 $: \[ P(0.49) = 0.49 \cdot (1 - 0.49) = 0.49 \cdot 0.51 = 0.2499 \]
For $ \hat{p} = 0.55 $: \[ P(0.55) = 0.55 \cdot (1 - 0.55) = 0.55 \cdot 0.45 = 0.2475 \]
For $ \hat{p} = 0.51 $: \[ P(0.51) = 0.51 \cdot (1 - 0.51) = 0.51 \cdot 0.49 = 0.2499 \]
For $ \hat{p} = 0.50 $: \[ P(0.50) = 0.50 \cdot (1 - 0.50) = 0.50 \cdot 0.50 = 0.25 \]

Step 3: Identify the Maximum Product

From the calculated values: