A household has weekly income of \(\$ 2000\). The mean weekly expenditure for households with this income is \(E(y \mid x=\$ 2000)=\mu_{y \mid x=\$ 2000}=\$ 220\), and expenditures exhibit variance \(\operatorname{var}(y \mid x=\$ 2,000)=\) \(\sigma_{y \mid x=\$ 2,000}^{2}=\$ 121\).

a. Assuming that weekly food expenditures are normally distributed, find the probability that a household with this income spends between \$200 and \$215 on food in a week. Include a sketch with your solution.

b. Find the probability that a household with this income spends more than \(\$ 250\) on food in a week. Include a sketch with your solution.

c. Find the probability in part (a) if the variance of weekly expenditures is \(\operatorname{var}(y \mid x=\$ 2,000)=\) \(\sigma_{y \mid x=\$ 2,000}^{2}=144\).

d. Find the probability in part (b) if the variance of weekly expenditures is \(\operatorname{var}(y \mid x=\$ 2,000)=\) \(\sigma_{y \mid x=\$ 2,000}^{2}=144\).