Abby Kratz, a market specialist at the market research firm of Saez, Sikes, and Spitz, is analyzing household budget data collected by her firm.Abby's dependent variable is weekly household expenditures on groceries (in $'s), and her independent variables are annual household income (in $1,000's)and household neighborhood (0 = suburban, 1 = rural).Regression analysis of the data yielded the following table. $$\begin{array} { | c | c | c | c | c | }
\hline & \text { Coefficients } & \text { Stardard Error } & t \text { Statistic } & p \text {-value } \
\hline \text { Intercept } & 19.68247 & 10.01176 & 1.965934 & 0.077667 \
\hline x _ { 1 } \text { (income) } & 1.735272 & 0.174564 & 9.940612 & 1.68 \mathrm { E } - 06 \
\hline x _ { 2 } \text { (neighborhood) } & 49.12456 & 7.655776 & 6.416667 & 7.67 \mathrm { E } - 05 \
\hline
\end{array}$$ However, Abby has reasons to believe that actually suburban households have a higher propensity to spend in groceries than rural households, as they tend to make more impulse purchasing decisions.In this new model, the actual income coefficient for the suburban families is 1.735272$\alpha$, for some $\alpha$ 1.If new data confirms that suburban houses with an annual income of $95,773.44 have the same weekly grocery spending as rural households, then $\alpha$ = ______.

Question

Abby Kratz, a market specialist at the market research firm of Saez, Sikes, and Spitz, is analyzing household budget data collected by her firm.Abby's dependent variable is weekly household expenditures on groceries (in $'s), and her independent variables are annual household income (in $1,000's)and household neighborhood (0 = suburban, 1 = rural).Regression analysis of the data yielded the following table. $$\begin{array} { | c | c | c | c | c | } 
\hline & \text { Coefficients } & \text { Stardard Error } & t \text { Statistic } & p \text {-value } \
\hline \text { Intercept } & 19.68247 & 10.01176 & 1.965934 & 0.077667 \
\hline x _ { 1 } \text { (income) } & 1.735272 & 0.174564 & 9.940612 & 1.68 \mathrm { E } - 06 \
\hline x _ { 2 } \text { (neighborhood) } & 49.12456 & 7.655776 & 6.416667 & 7.67 \mathrm { E } - 05 \
\hline
\end{array}$$ However, Abby has reasons to believe that actually suburban households have a higher propensity to spend in groceries than rural households, as they tend to make more impulse purchasing decisions.In this new model, the actual income coefficient for the suburban families is 1.735272$\alpha$, for some $\alpha$ > 1.If new data confirms that suburban houses with an annual income of $95,773.44 have the same weekly grocery spending as rural households, then $\alpha$ = ______.

Quizplus · Accepted Answer

Given that suburban and rural households with an annual income of $95,773.44 spend the same amount on groceries weekly, we can set up an equation to solve for α. Let's denote the annual income as I = $95,773.44, which is 95.77344 in thousands of dollars. The equation for the suburban household's grocery spending (S) and the rural household's grocery spending (R) can be equated as follows:$$S = 19.68247 + (1.735272\alpha \times 95.77344) + 49.12456 \times 0$$$$R = 19.68247 + (1.735272 \times 95.77344) + 49.12456 \times 1$$Setting S equal to R and solving for α gives us:$$19.68247 + (1.735272\alpha \times 95.77344) = 19.68247 + (1.735272 \times 95.77344) + 49.12456$$Simplifying and solving for α:$$1.735272\alpha \times 95.77344 = 1.735272 \times 95.77344 + 49.12456$$$$1.735272\alpha = 1.735272 + \frac{49.12456}{95.77344}$$$$1.735272\alpha = 1.735272 + 0.512815$$$$1.735272\alpha = 2.248087$$$$\alpha = \frac{2.248087}{1.735272}$$$$\alpha = 1.295587$$Therefore, the correct value of α is 1.295587.

Abby Kratz, a Market Specialist at the Market Research Firm