Question
The table compares the average daily temperature and ice cream sales each day.
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
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
Answers
GPT-4o mini
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**
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
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**