Questions: Complete the tables for different values of p̂ and q̂=1−p̂. From the tables, which value of p̂ appears to give the maximum value of the product p̂q̂. Complete the table below. p̂ q̂=1−p̂ p̂q̂ 0.0 1.0 0.00 0.1 0.9 0.09 0.2 0.8 0.3 0.4 0.5 (Type integers or decimals.)

Complete the tables for different values of p̂ and q̂=1−p̂. From the tables, which value of p̂ appears to give the maximum value of the product p̂q̂.

Complete the table below.
p̂ q̂=1−p̂ p̂q̂
0.0 1.0 0.00
0.1 0.9 0.09
0.2 0.8
0.3
0.4
0.5
(Type integers or decimals.)

Transcript text: Complete the tables for different values of $\hat{p}$ and $\hat{q}=1-\hat{p}$. From the tables, which value of $\hat{p}$ appears to give the maximum value of the product $\hat{p q}$. Complete the table below. \begin{tabular}{|c|c|c|} \hline$\hat{\mathbf{p}}$ & $\hat{\mathbf{q}}=\mathbf{1}-\hat{\mathbf{p}}$ & $\hat{\mathbf{p q}}$ \\ \hline 0.0 & 1.0 & 0.00 \\ \hline 0.1 & 0.9 & 0.09 \\ \hline 0.2 & 0.8 & $\square$ \\ \hline 0.3 & $\square$ & $\square$ \\ \hline 0.4 & $\square$ & $\square$ \\ \hline 0.5 & $\square$ & $\square$ \\ \hline \end{tabular} (Type integers or decimals.)

Solution

Solution Steps

To complete the table, we need to calculate the values of $\hat{q}$ using the formula $\hat{q} = 1 - \hat{p}$ for each given $\hat{p}$. Then, compute the product $\hat{p q}$ for each pair of $\hat{p}$ and $\hat{q}$. Finally, identify the value of $\hat{p}$ that gives the maximum value of the product $\hat{p q}$.

Step 1: Calculate $ \hat{q} $ Values

For each value of $ \hat{p} $, we calculate $ \hat{q} $ using the formula: \[ \hat{q} = 1 - \hat{p} \] The calculated values are:

For $ \hat{p} = 0.0 $, $ \hat{q} = 1.0 $
For $ \hat{p} = 0.1 $, $ \hat{q} = 0.9 $
For $ \hat{p} = 0.2 $, $ \hat{q} = 0.8 $
For $ \hat{p} = 0.3 $, $ \hat{q} = 0.7 $
For $ \hat{p} = 0.4 $, $ \hat{q} = 0.6 $
For $ \hat{p} = 0.5 $, $ \hat{q} = 0.5 $

Step 2: Calculate $ \hat{p q} $ Values

Next, we compute the product $ \hat{p q} $ for each pair of $ \hat{p} $ and $ \hat{q} $: \[ \hat{p q} = \hat{p} \cdot \hat{q} \] The calculated products are: