Question

is there a linear relationship between average daily temperature (°f) and number of coffees sold at dash grind coffee shop? below is a random sample of 11 days in a year.
temperature (°f) 45 60 75 28 40 65 85 95 50 70 95
number hot coffees sold 38 17 10 58 51 35 24 16 44 29 15
a) what is the explanatory variable?
what is the response variable?
b) what is the least - squares regression equation?
c) what is the correlation coefficient, r? interpret this result.
d) what is the coefficient of determination, r²? interpret this result.
e) use the equation in part (b) to estimate the average number of hot coffees sold if the temperature is 67°f.

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) $42.82$