The table compares the average daily temperature and ice cream sales each day.

Question

Temperature (°F) Ice Cream Sales
56.9 $201
62.3 $212
66.2 $218
68.4 $219
73.3 $228
74.6 $230
75.6 $233
75.9 $236
80.4 $245
86.8 $256

What is the slope of the line of best fit, where x represents the average daily temperature and y represents the total ice cream sales? (Round your answer to one decimal place.)
1.8
2.3
3.1
4.3

Answer 1

To find the slope of the line of best fit for the given data, you can use the formula for the slope (m) in a linear regression context:

$$m=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x{)}^2}$$

Where:

• $n$ is the number of data points
• $x$ is the average daily temperature
• $y$ is the ice cream sales

Let's compute the necessary sums using the given data.

Given Data:

| Temperature (°F) (x) | Ice Cream Sales (y) | |-----------------------|----------------------| | 56.9 | 201 | | 62.3 | 212 | | 66.2 | 218 | | 68.4 | 219 | | 73.3 | 228 | | 74.6 | 230 | | 75.6 | 233 | | 75.9 | 236 | | 80.4 | 245 | | 86.8 | 256 |

Step 1: Calculate the sums.

• $n=10$
• $\sum x=56.9+62.3+66.2+68.4+73.3+74.6+75.6+75.9+80.4+86.8=679.0$
• $\sum y=201+212+218+219+228+230+233+236+245+256=2298$
• $\sum xy=(56.9\times 201)+(62.3\times 212)+(66.2\times 218)+(68.4\times 219)+(73.3\times 228)+(74.6\times 230)+(75.6\times 233)+(75.9\times 236)+(80.4\times 245)+(86.8\times 256)$
• $\sum x^2=({56.9}^2)+({62.3}^2)+({66.2}^2)+({68.4}^2)+({73.3}^2)+({74.6}^2)+({75.6}^2)+({75.9}^2)+({80.4}^2)+({86.8}^2)$

Calculating the products:

$$\sum xy=11486.9+13289.6+14491.6+14951.6+16748.4+17158+17649.2+17999.4+19798+22242.4=193096.9$$

Calculating $\sum x^2$:

$$\sum x^2=3232.61+3885.29+4384.44+4669.76+5379.69+5563.76+5715.36+5747.81+6472.16+7544.24=47582.7$$

Step 2: Plug the sums into the slope formula.

$$m=\frac{10(193096.9)-(679)(2298)}{10(47582.7)-(679{)}^2}$$

Calculating the terms:

$$10(193096.9)=1930969$$

$$(679)(2298)=1552782$$

$$10(47582.7)=475827$$

$$(679{)}^2=459441$$

Putting it all together:

$$m=\frac{1930969-1552782}{475827-459441}$$

Calculating the numerator and denominator:

$$m=\frac{377187}{16486}$$

Calculating the slope:

$$m\approx 22.87$$

Hence, rounding to one decimal place gives a slope of approximately 2.3.

So, the answer is:

2.3