A restaurant manager records the number of tables seated $(x)$ and the total number of cups of complimentary salsa $(y)$ served, as shown in the tables below. Which statements show correct values for the median-fit method? Check all that apply.
| x | y |
| --- | ------ |
| 35 | 99.5 |
| 38 | 97.25 |
| 41 | 102 |
| 47 | 110.5 |
| 52 | 103.75 |
| x | y |
| --- | --- |
| 55 | 125 |
| 58 | 160 |
| 63 | 142 |
| 71 | 153 |
A. The left summary point is $(38,97.25)$.
B. The middle summary point is $(52,110.5)$.
C. The right summary point is $(63,160)$.
D. The slope of the line of best fit is 2.14.
E. The $y$-intercept of the line of best fit is -10.48.
Answer (2)
Calculate the median x and y values for the left, middle, and right groups to find the summary points: (38, 99.5), (52, 110.5), and (63, 153).
Calculate the slope using the left and right summary points: m = 63 − 38 153 − 99.5 = 2.14 .
Calculate the y-intercept using the slope and the middle summary point: b = 110.5 − 2.14 ∗ 52 = − 0.78 .
Compare the calculated values with the given statements to determine the correct ones: The middle summary point is (52, 110.5) and the slope of the line of best fit is 2.14.
Explanation
Finding the Median Values First, we need to find the median of the x and y values for the left, middle, and right groups. The data is already divided into these groups:
Left group: x values: 35, 38, 41 y values: 99.5, 97.25, 102
Middle group: x values: 47, 52, 55 y values: 110.5, 103.75, 125
Right group: x values: 58, 63, 71 y values: 160, 142, 153
Calculating Summary Points Now, let's calculate the median for each group:
Left group: Median x = 38 Median y = 99.5 Left summary point: (38, 99.5)
Middle group: Median x = 52 Median y = 110.5 Middle summary point: (52, 110.5)
Right group: Median x = 63 Median y = 153 Right summary point: (63, 153)
Calculating the Slope Next, we calculate the slope (m) using the left ( x 1 , y 1 ) and right ( x 2 , y 2 ) summary points:
m = x 2 − x 1 y 2 − y 1 = 63 − 38 153 − 99.5 = 25 53.5 = 2.14
Calculating the y-intercept Now, we calculate the y-intercept (b) using the slope (m) and the middle summary point ( x m , y m ) :
b = y m − m ∗ x m = 110.5 − 2.14 ∗ 52 = 110.5 − 111.28 = − 0.78
Comparing Calculated Values with Given Statements Comparing the calculated values with the given statements:
The left summary point is (38, 97.25). Our calculated left summary point is (38, 99.5). This statement is incorrect. The middle summary point is (52, 110.5). Our calculated middle summary point is (52, 110.5). This statement is correct. The right summary point is (63, 160). Our calculated right summary point is (63, 153). This statement is incorrect. The slope of the line of best fit is 2.14. Our calculated slope is 2.14. This statement is correct. The y-intercept of the line of best fit is -10.48. Our calculated y-intercept is -0.78. This statement is incorrect.
Final Answer Therefore, the correct statements are:
The middle summary point is (52, 110.5). The slope of the line of best fit is 2.14.
Examples
Understanding the relationship between variables is crucial in many real-world scenarios. For instance, a marketing team might track the number of ads displayed (x) and the resulting sales (y). By using the median-fit method, they can determine the line of best fit, which helps them predict future sales based on ad spend. This method is also applicable in environmental science, where researchers might analyze the correlation between pollution levels (x) and the health of a local ecosystem (y), enabling them to make informed decisions about environmental policies. The median-fit method provides a robust way to model relationships and make predictions in various fields.
The correct statements about the summary points and line of best fit are: the middle summary point is (52, 110.5) and the slope of the line of best fit is 2.14. Statements A, C, and E are incorrect. So, the answer choices are B and D.
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