Question

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 $hat{p}hat{q}$.

complete the table below
(type integers or decimals )

Explanation:

Step1: Calculate $\hat{p}\hat{q}$ for $\hat{p}=0.2$

$\hat{p}=0.2$, $\hat{q}=0.8$, so $\hat{p}\hat{q}=0.2\times0.8=0.16$

Step2: Compute $\hat{q}$ for $\hat{p}=0.3$

$\hat{q}=1-\hat{p}=1-0.3=0.7$

Step3: Calculate $\hat{p}\hat{q}$ for $\hat{p}=0.3$

$\hat{p}\hat{q}=0.3\times0.7=0.21$

Step4: Compute $\hat{q}$ for $\hat{p}=0.4$

$\hat{q}=1-\hat{p}=1-0.4=0.6$

Step5: Calculate $\hat{p}\hat{q}$ for $\hat{p}=0.4$

$\hat{p}\hat{q}=0.4\times0.6=0.24$

Step6: Compute $\hat{q}$ for $\hat{p}=0.5$

$\hat{q}=1-\hat{p}=1-0.5=0.5$

Step7: Calculate $\hat{p}\hat{q}$ for $\hat{p}=0.5$

$\hat{p}\hat{q}=0.5\times0.5=0.25$

Step8: Identify max $\hat{p}\hat{q}$ value

Compare all products: $0.00, 0.09, 0.16, 0.21, 0.24, 0.25$. The maximum is $0.25$ at $\hat{p}=0.5$.

Answer:

Completed Table:

$\hat{p}$	$\hat{q}=1-\hat{p}$	$\hat{p}\hat{q}$
0.1	0.9	0.09
0.2	0.8	0.16
0.3	0.7	0.21
0.4	0.6	0.24
0.5	0.5	0.25

Maximum Product:

The value $\hat{p}=0.5$ gives the maximum value of $\hat{p}\hat{q}$ (which is 0.25).