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
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