Melinda works at a cafe. Each day that she works, she records $x$, the total dollar amount of her customers' bills and then $y$, her total daily wages. The table shows her data for 2 weeks.
| | |
| --- | --- |
| x | y |
| 50 | 36 |
| 100 | 43 |
| 75 | 38 |
| 80 | 40 |
| 90 | 42 |
| 140 | 50 |
| 150 | 60 |
| 95 | 43 |
| 125 | 46 |
| 160 | 50 |
| 165 | 55 |
According to the line of best fit, what is the minimum amount, to the nearest dollar, Melinda will earn for each day of work, even if she serves no customers?
A. $18
B. $26
C. $36
D. $40
Answer (2)
Calculate the slope m of the line of best fit using the formula m = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y ) , which results in m ≈ 0.177 .
Calculate the y-intercept b of the line of best fit using the formula b = n ( ∑ y ) − m ( ∑ x ) , which results in b ≈ 25.936 .
Round the y-intercept b to the nearest dollar, which gives 26 .
The minimum amount Melinda will earn each day is 26 .
Explanation
Understanding the Problem We are given a set of data points relating Melinda's daily wages ( y ) to the total dollar amount of her customers' bills ( x ). We need to find the minimum amount Melinda earns each day, which corresponds to the y-intercept of the line of best fit. This represents her earnings when she serves no customers ( x = 0 ).
Finding the Line of Best Fit The line of best fit is represented by the equation y = m x + b , where m is the slope and b is the y-intercept. We need to calculate m and b using the given data.
Calculating the Slope First, let's calculate the slope m using the formula: m = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y ) where n is the number of data points. We have the following data: x = [ 50 , 100 , 75 , 80 , 90 , 140 , 150 , 95 , 125 , 160 , 165 ] y = [ 36 , 43 , 38 , 40 , 42 , 50 , 60 , 43 , 46 , 50 , 55 ] Using the python_calculation_tool, we find that m ≈ 0.177 .
Calculating the Y-Intercept Next, we calculate the y-intercept b using the formula: b = n ( ∑ y ) − m ( ∑ x ) Using the python_calculation_tool, we find that b ≈ 25.936 .
Rounding to the Nearest Dollar We need to round the y-intercept b to the nearest dollar. Since b ≈ 25.936 , rounding to the nearest dollar gives us 26 .
Final Answer Therefore, the minimum amount Melinda will earn for each day of work, even if she serves no customers, is approximately $26 .
Examples
Understanding the line of best fit can help in various real-life scenarios. For example, if you are tracking your study hours ( x ) and your exam scores ( y ), the line of best fit can help you predict your score based on the number of hours you study. The y-intercept would represent your expected score even if you didn't study at all, which could be due to prior knowledge or innate ability. This kind of analysis is also used in business to predict sales based on advertising expenditure or to forecast production costs based on the number of units produced. The equation of the line of best fit is a powerful tool for making predictions and understanding relationships between variables.
Melinda's minimum daily earnings when serving no customers is approximately $26, which corresponds to the y-intercept of the line of best fit. This value represents her earnings when the total dollar amount of customers' bills is zero. The calculations confirm that the correct answer is $26 (option B).
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