1. **State the problem:** We have data on the number of T-shirts ordered and the cost per shirt. We want to find a linear regression equation relating order size ($x$) to cost per shirt ($y$), interpret the slope and intercept, and use the equation to predict costs for given order sizes. 2. **Data points:** $(500, 3.25), (700, 1.95), (200, 5.20), (460, 3.51), (740, 1.69)$. 3. **Linear regression formula:** The line is $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. 4. **Calculate slope ($m$) and intercept ($b$):** Using technology or formulas, $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ where $n=5$. 5. **Calculate sums:** $\sum x = 500 + 700 + 200 + 460 + 740 = 2600$ $\sum y = 3.25 + 1.95 + 5.20 + 3.51 + 1.69 = 15.6$ $\sum xy = 500\times3.25 + 700\times1.95 + 200\times5.20 + 460\times3.51 + 740\times1.69 = 1625 + 1365 + 1040 + 1614.6 + 1250.6 = 6895.2$ $\sum x^2 = 500^2 + 700^2 + 200^2 + 460^2 + 740^2 = 250000 + 490000 + 40000 + 211600 + 547600 = 1,539,200$ 6. **Calculate slope:** $$m = \frac{5 \times 6895.2 - 2600 \times 15.6}{5 \times 1,539,200 - 2600^2} = \frac{34476 - 40560}{7,696,000 - 6,760,000} = \frac{-6084}{936,000} \approx -0.0065$$ 7. **Calculate intercept:** $$b = \frac{15.6 - (-0.0065) \times 2600}{5} = \frac{15.6 + 16.9}{5} = \frac{32.5}{5} = 6.5$$ 8. **Linear regression equation:** $$y = -0.0065x + 6.5$$ 9. **Interpret slope and intercept:** - Slope $-0.0065$ means for each additional shirt ordered, the cost per shirt decreases by about 0.0065. - Intercept $6.5$ is the estimated cost per shirt if zero shirts were ordered (theoretical starting cost). 10. **Extrapolate order size for price $1.50$:** Set $y=1.50$: $$1.50 = -0.0065x + 6.5$$ $$-0.0065x = 1.50 - 6.5 = -5.0$$ $$x = \frac{-5.0}{-0.0065} \approx 769.23$$ So about 770 shirts are needed to get $1.50$ per shirt. 11. **Interpolate cost for 350 shirts:** $$y = -0.0065 \times 350 + 6.5 = -2.275 + 6.5 = 4.225$$ Cost per shirt is approximately 4.23 for 350 shirts. **Final answers:** - Linear regression equation: $y = -0.0065x + 6.5$ - Slope meaning: cost decreases by 0.0065 per additional shirt. - Intercept meaning: cost per shirt if no shirts ordered is 6.5. - Order size for $1.50$ per shirt: about 770 shirts. - Cost for 350 shirts: about 4.23 per shirt.