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The mean is $\mathrm{μ}=250$and standard deviation is $\mathrm{σ}=5000$

The number of homeowners is $\mathrm{n}=1000$

The sample mean $\stackrel{¯}{\mathrm{x}}=300$

The following concept was used:

$\mathrm{z}=\frac{\mathrm{x}−{\mathrm{μ}}_{\stackrel{¯}{\mathrm{x}}}}{{\mathrm{σ}}_{\stackrel{¯}{\mathrm{x}}}}$

The sampling distribution of the sample mean $\stackrel{¯}{\mathrm{x}}$is also normal because the population distribution is normal.

The Z-score is

Using the normal probability, the associating probability is calculated.

$\mathrm{P}(\mathrm{Z}<0.32)$s typical normal probability table in the appendix has a row beginning with $0.3$ and a column beginning with $0.2.$

$\begin{array}{rcl}P(\stackrel{¯}{X}& â‰\yen & 300)=P(Z>0.32)\\ & =& 1-P(Z<0.32)\\ & =& 1-0.6255\\ & =& 0.3745\\ & =& 37.45\%\end{array}$

The mean is $\mathrm{μ}=250$and standard deviation is $\mathrm{σ}=5000$

The number of homeowners is $\mathrm{n}=1000$

$\mathrm{P}(\stackrel{¯}{\mathrm{X}}≤\stackrel{¯}{\mathrm{x}})=90\mathrm{\%}$

The following concept was used:

$\mathrm{z}=\frac{\mathrm{x}−{\mathrm{μ}}_{\stackrel{¯}{\mathrm{x}}}}{{\mathrm{σ}}_{\stackrel{¯}{\mathrm{x}}}}$

Determine the z-score in the normal probability table that corresponds to a probability of $90$percent or $0.90$, and the closest probability is width="51">0.8997, which is located in the normal probability table's row $1.2$ and column$.08$, and hence the equivalent z-score is $1.2+.08=1.28$.

localid="1657629267213" $\mathrm{z}=1.28$

Z-score is localid="1657629270630" $\mathrm{z}=\frac{\mathrm{x}−{\mathrm{μ}}_{\stackrel{¯}{\mathrm{x}}}}{{\mathrm{σ}}_{\stackrel{¯}{\mathrm{x}}}}=\frac{\stackrel{¯}{\mathrm{x}}−\mathrm{μ}}{\mathrm{σ}/\sqrt{\mathrm{n}}}=\frac{\stackrel{¯}{\mathrm{x}}−250}{5000\sqrt{1000}}$

The found expressions of the z-two score must then be equal:

localid="1657629274503" $\begin{array}{r}\frac{\stackrel{¯}{\mathrm{x}}−250}{5000/\sqrt{1000}}=1.28\\ \stackrel{¯}{\mathrm{x}}−250=1.28(5000/\sqrt{1000})\end{array}$

To each side, add $250$

localid="1657629377013" $\stackrel{¯}{\mathrm{x}}−250+1.28(5000/\sqrt{1000})$

Determine:

localid="1657629382095" $\mathrm{x}=452.39$

As a result, the business should charge is localid="1657629389088" $\mathrm{\$}452.39.$