$\backslash$table$[[,,,$Income $($$$),$Size$],[,1,4\mathrm{.}2,73\mathrm{.}98,3],[,2,4\mathrm{.}08,54\mathrm{.}9,2],[,3,5\mathrm{.}76,90\mathrm{.}67,4],[,4,3\mathrm{.}48,52\mathrm{.}02,1],[,5,4\mathrm{.}2,65\mathrm{.}7,2],[,6,4\mathrm{.}8,53\mathrm{.}64,4],[,7,4\mathrm{.}32,79\mathrm{.}74,3],[,8,5\mathrm{.}04,69\mathrm{.}58,4],[10,9,6\mathrm{.}12,165\mathrm{.}6,5],[,10,3\mathrm{.}24,64\mathrm{.}8,1],[1,11,4\mathrm{.}8,138\mathrm{.}42,3],[1,12,3\mathrm{.}24,125\mathrm{.}82,1],[,13,5\mathrm{.}67,77\mathrm{.}52,7],[15,14,4\mathrm{.}32,100\mathrm{.}16,4],[,15,6\mathrm{.}6,169\mathrm{.}39,8],[,16,5\mathrm{.}4,141\mathrm{.}36,3],[,17,6,36\mathrm{.}96,5],[,18,5\mathrm{.}4,56\mathrm{.}88,4],[,19,3\mathrm{.}63,71\mathrm{.}82,1.],[,20,4\mathrm{.}86,69\mathrm{.}43,3],[,21,4\mathrm{.}23,54\mathrm{.}36,2],[,22,5\mathrm{.}25,87\mathrm{.}65,5],[,23,4\mathrm{.}65,38\mathrm{.}16,3],[,24,5\mathrm{.}4,43\mathrm{.}74,7],[,25,8\mathrm{.}2,48\mathrm{.}82,8]]$

Many regions in North and South Carolina and Georgia have experienced rapid population growth over the last $10$ years. It is expected that the growth will continue over the next $10$ years. This has motivated many of the large grocery store chains to build new stores in the region. The Kelley's Super Grocery Stores Inc. chain is no exception. The director of planning for Kelley's Super Grocery Stores wants to study adding more stores in this region. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family The director collected Grocery Expenditure $($in $$1000),$ family income $($in $$1000),$ and family size for a random sample of $25$ families in the areas. Data can be found in file INCOME.

a$.$ Run a multiple linear regression and report the estimated regression line.

b$.$ Perform the appropriate test to determine if the model is valid.

c$.$ Interpret the coefficient for family income.