A restaurant manager records the number of tables seated $(x)$ and the total number of cups of complimentary salsa $(y)$ served. Which statements show correct values for the median-fit method? Check all that apply.
The left summary point is $(38,97.25)$.
The middle summary point is $(52,110.5)$.
The right summary point is $(63,160)$.
The slope of the line of best fit is 2.14.
The $y$-intercept of the line of best fit is -10.48.
Answer (2)
Divide the data into left, middle, and right groups based on x-values.
Calculate the median x and y values for each group to find the summary points: (38, 99.5), (49.5, 107.125), and (60.5, 147.5).
Determine the slope using the left and right summary points: m ≈ 2.13 .
Calculate the y-intercept using the slope and the middle summary point: b ≈ 1.53 .
Compare the calculated values with the given statements and find that none of them are correct. Therefore, no statements are correct.
Explanation
Understanding the Problem We are given a data set of tables seated ( x ) versus cups of salsa served ( y ) and asked to evaluate statements about the median-fit line. The median-fit method involves dividing the data into three groups, finding the median x and y values for each group to create summary points, and then using these points to determine the slope and y -intercept of the line of best fit.
Dividing the Data First, we need to divide the data into three groups: left, middle, and right, based on the x values.
Left group: (35, 99.5), (38, 97.25), (41, 102) Middle group: (47, 110.5), (52, 103.75) Right group: (55, 125), (58, 160), (63, 142), (71, 153)
Finding the Summary Points Next, we find the median x and y values for each group to determine the summary points.
Left summary point: The median of the x values [35, 38, 41] is 38, and the median of the y values [99.5, 97.25, 102] is 99.5. So, the left summary point is (38, 99.5).
Middle summary point: The median of the x values [47, 52] is (47+52)/2 = 49.5, and the median of the y values [110.5, 103.75] is (110.5+103.75)/2 = 107.125. So the middle summary point is (49.5, 107.125).
Right summary point: The median of the x values [55, 58, 63, 71] is (58+63)/2 = 60.5, and the median of the y values [125, 160, 142, 153] is (142+153)/2 = 147.5. So the right summary point is (60.5, 147.5).
Calculating the Slope Now, we calculate the slope ( m ) using the left and right summary points: ( x 1 , y 1 ) = ( 38 , 99.5 ) and ( x 2 , y 2 ) = ( 60.5 , 147.5 ) .
m = x 2 − x 1 y 2 − y 1 = 60.5 − 38 147.5 − 99.5 = 22.5 48 = 2.13333... ≈ 2.13
Calculating the y-intercept Next, we calculate the y -intercept ( b ) using the slope and the middle summary point: ( x , y ) = ( 49.5 , 107.125 ) .
b = y − m x = 107.125 − 2.13333... × 49.5 = 107.125 − 105.599835 ≈ 1.53
Comparing with Given Statements Finally, we compare the calculated summary points, slope, and y -intercept with the given statements to determine which statements are correct.
The left summary point is (38, 97.25). Our calculation is (38, 99.5). So, this statement is incorrect. The middle summary point is (52, 110.5). Our calculation is (49.5, 107.125). So, this statement is incorrect. The right summary point is (63, 160). Our calculation is (60.5, 147.5). So, this statement is incorrect. The slope of the line of best fit is 2.14. Our calculation is approximately 2.13. So, this statement is incorrect. The y -intercept of the line of best fit is -10.48. Our calculation is approximately 1.53. So, this statement is incorrect.
Examples
Imagine you're managing a coffee shop and want to understand the relationship between the number of customers and the amount of coffee beans you use. By tracking these two variables and using the median-fit method, you can create a line of best fit. This line helps you predict how many coffee beans you'll need based on the number of customers you expect, allowing you to manage your inventory more efficiently. This method is also applicable in scenarios like predicting sales based on advertising spend or estimating resource needs based on project size, providing a simple yet effective way to make data-driven decisions.
All the provided statements regarding the median-fit method are incorrect based on calculated summary points, slope, and y-intercept.
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