is there a linear relationship between average daily temperature (°f) and number of coffees sold at dash…

Explanation:

Step1: Identify variables

The explanatory variable is the one that may influence the other. Here, the average - daily temperature (in °F) influences the number of hot coffees sold. The response variable is the one being influenced. So, the explanatory variable is temperature ($x$) and the response variable is the number of hot coffees sold ($y$).

Step2: Calculate regression equation

Let $n = 11$. First, calculate $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, $\sum xy$. Let $x$ be the temperature and $y$ be the number of hot - coffees sold. $\sum x=45 + 60+75+78+40+65+80+45+50+70+95 = 673$ $\sum y=58 + 37+40+58+51+33+24+56+44+29+45 = 475$ $\sum x^2=45^2+60^2+75^2+78^2+40^2+65^2+80^2+45^2+50^2+70^2+95^2$ $=2025 + 3600+5625+6084+1600+4225+6400+2025+2500+4900+9025 = 48034$ $\sum y^2=58^2+37^2+40^2+58^2+51^2+33^2+24^2+56^2+44^2+29^2+45^2$ $=3364+1369+1600+3364+2601+1089+576+3136+1936+841+2025 = 21801$ $\sum xy=(45\times58)+(60\times37)+(75\times40)+(78\times58)+(40\times51)+(65\times33)+(80\times24)+(45\times56)+(50\times44)+(70\times29)+(95\times45)$ $=2610+2220+3000+4524+2040+2145+1920+2520+2200+2030+4275 = 29484$

The slope $b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2}$ $b_1=\frac{11\times29484-673\times475}{11\times48034-(673)^2}$ $=\frac{324324 - 319675}{528374 - 452929}=\frac{4649}{75445}\approx - 0.062$

The intercept $b_0=\bar{y}-b_1\bar{x}$, where $\bar{x}=\frac{\sum x}{n}=\frac{673}{11}\approx61.18$ and $\bar{y}=\frac{\sum y}{n}=\frac{475}{11}\approx43.18$ $b_0 = 43.18-(-0.062)\times61.18=43.18 + 3.793 = 46.973$ The least - squares regression equation is $\hat{y}=46.973-0.062x$

Step3: Calculate correlation coefficient

The correlation coefficient $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$ $r=\frac{11\times29484-673\times475}{\sqrt{(11\times48034-(673)^2)(11\times21801-(475)^2)}}$ $=\frac{4649}{\sqrt{75445\times44736}}\approx - 0.89$ Interpretation: The value of $r\approx - 0.89$ indicates a strong negative linear relationship between temperature and the number of hot coffees sold. As the temperature increases, the number of hot coffees sold tends to decrease.

Step4: Calculate coefficient of determination

The coefficient of determination $r^2=(-0.89)^2 = 0.7921$ Interpretation: Approximately $79.21%$ of the variation in the number of hot coffees sold can be explained by the linear relationship with temperature.

Step5: Make prediction

Substitute $x = 67$ into the regression equation $\hat{y}=46.973-0.062x$ $\hat{y}=46.973-0.062\times67$ $\hat{y}=46.973 - 4.154=42.819\approx42.82$

Answer:

a) Explanatory variable: Temperature; Response variable: Number of hot coffees sold b) $\hat{y}=46.973 - 0.062x$ c) $r\approx - 0.89$. There is a strong negative linear relationship between temperature and the number of hot coffees sold. d) $r^2 = 0.7921$. Approximately $79.21%$ of the variation in the number of hot coffees sold can be explained by the linear relationship with temperature. e) $42.82$