the table shows data from a party planner, representing the number of people at an event (x) and the total…

Explanation:

Step1: Find the slope between two points

First, we find the slope between ((75, 2100)) and ((140, 3100)). The slope formula is (m=\frac{y_2 - y_1}{x_2 - x_1}). So, (m_1=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.38). Then, find the slope between ((140, 3100)) and ((200, 4200)). (m_2=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.33). The median slope is the median of (m_1) and (m_2). Arranging (15.38) and (18.33) in order, the median is (\frac{15.38 + 18.33}{2}=\frac{33.71}{2}\approx16.86). We can also use two points to find the slope more accurately. Let's use ((75, 2100)) and ((200, 4200)) for a better approximation. (m=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8).

Step2: Use the slope-intercept form

The slope-intercept form is (y = mx + b), where (b) is the y-intercept. We can use one of the points, say ((75, 2100)) and (m = 16.8). Plugging into the formula: (2100=16.8\times75 + b). Calculate (16.8\times75 = 1260). Then, (b=2100 - 1260 = 840)? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's use the median - fit method properly. The median - fit method uses three summary points: the first is the median of the (x) - values of the first half, the second is the median of all (x) - values, and the third is the median of the (x) - values of the second half. But here we are given the summary points ((75,2100)), ((140,3100)), and ((200,4200)). Let's find the equation of the line passing through two of these points and then find the y - intercept. Let's use ((75,2100)) and ((200,4200)). The slope (m=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8). Now, using the point - slope form (y - y_1=m(x - x_1)) with ((x_1,y_1)=(75,2100)): (y-2100 = 16.8(x - 75)). To find the y - intercept, set (x = 0): (y=16.8\times(- 75)+2100). Calculate (16.8\times(-75)=-1260). Then (y=-1260 + 2100=840)? Wait, but let's check with another point. Let's use ((140,3100)). (y=16.8x + b), so (3100=16.8\times140 + b). (16.8\times140 = 2352). Then (b=3100 - 2352 = 748)? Wait, there is a discrepancy. Wait, maybe my choice of points is wrong. Let's use the median - fit method steps. The median - fit line is determined by first finding the median of the (x) - values (the middle (x) - value, which is the 5th value? Wait, the (x) - values are 50,75,100,125,140,150,175,200,220? Wait, no, the table has (x) values: 50,75,100,125,140,150,175,200,220? Wait, the number of data points for (x) is 9? Wait, the (x) values are 50,75,100,125,140,150,175,200,220 (9 points). The median of (x) is the 5th value, which is 140. Then the first half (excluding the median) is the first 4 points: 50,75,100,125, median of first half is 75 (average of 75 and 100? Wait, no, for even number of points in the first half (4 points: 50,75,100,125), the median is (\frac{75 + 100}{2}=87.5), but the problem gives the summary points as (75,2100), (140,3100), (200,4200). So we can use these three points. Let's find the equation of the line that best fits these three points. We can use linear regression on these three points. The general formula for the line through three points can be found by solving the system. Let the line be (y=mx + b). For ((75,2100)): (2100 = 75m + b) (1). For ((140,3100)): (3100=140m + b) (2). For ((200,4200)): (4200 = 200m + b) (3). Subtract (1) from (2): (3100 - 2100=(140m + b)-(75m + b)) => (1000 = 65m) => (m=\frac{1000}{65}\approx15.38). Subtract (2) from (3): (4200 - 3100=(200m + b)-(140m + b)) => (1100 = 60m) => (m=\frac{1100}{60}\approx18.33). The median slope is the median of (15.38) and (18.33), which is (\frac{15.38 + 18.33}{2}\approx16.86). Now, we can use the average of the three (b) values from the three equations. From (1): (b = 2100-75m). From (2): (b = 3100 - 140m). From (3): (b=4200 - 200m). Let's use (m = 16.86). From (1): (b=2100-75\times16.86=2100 - 1264.5 = 835.5). From (2): (b=3100-140\times16.86=3100 - 2360.4 = 739.6). From (3): (b=4200-200\times16.86=4200 - 3372 = 828). The median of (835.5), (739.6), and (828) is (828)? Wait, no, that's not right. Wait, maybe a better way is to use two points to find the line. Let's use (75,2100) and (200,4200). The slope (m=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8). Now, using the point - slope form with (75,2100): (y - 2100=16.8(x - 75)). To find the y - intercept, set (x = 0): (y=16.8\times(-75)+2100=-1260 + 2100 = 840)? But wait, when we use (140,3100) with (m = 16.8): (y=16.8\times140 + b) => (3100=2352 + b) => (b=3100 - 2352 = 748). There is a conflict. Wait, the median - fit method uses three lines: one through the first summary point and the median summary point, one through the median summary point and the third summary point, and the median - fit line is the median of these three lines' slopes and y - intercepts. Wait, the three summary points are (75,2100) [first half median], (140,3100) [overall median], (200,4200) [second half median]. We can find the equation of the line through (75,2100) and (140,3100): slope (m_1=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.38), equation: (y - 2100=15.38(x - 75)), so (y=15.38x-15.38\times75 + 2100=15.38x-1153.5 + 2100=15.38x + 946.5). The equation of the line through (140,3100) and (200,4200): slope (m_2=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.33), equation: (y - 3100=18.33(x - 140)), so (y=18.33x-18.33\times140 + 3100=18.33x-2566.2 + 3100=18.33x + 533.8). The equation of the line through (75,2100) and (200,4200): slope (m_3 = 16.8), equation: (y=16.8x + 840) (as before). Now, we need to find the median of the three y - intercepts: 946.5, 533.8, and 840. Arranging them in order: 533.8, 840, 946.5. The median is 840? No, wait, 533.8, 840, 946.5: the middle one is 840. But when we calculated using (140,3100) and (m = 16.8), we got (b = 748). Wait, maybe I made a mistake in the slope. Let's recalculate the slope between (75,2100) and (140,3100): (m=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.38). Then the equation is (y=15.38x + b). Using (75,2100): (2100=15.38\times75 + b) => (2100 = 1153.5 + b) => (b=2100 - 1153.5 = 946.5). Between (140,3100) and (200,4200): (m=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.33), equation: (y=18.33x + b), using (140,3100): (3100=18.33\times140 + b) => (3100 = 2566.2 + b) => (b=3100 - 2566.2 = 533.8). Between (75,2100) and (200,4200): (m = 16.8), (b = 840). Now, the three y - intercepts are 946.5, 533.8, and 840. Arranging them in ascending order: 533.8, 840, 946.5. The median of these three is 840? No, wait, 533.8, 840, 946.5: the median is 840. But the option 748 is there. Wait, maybe I used the wrong points. Wait, the problem says "summary points (75, 2,100), (140, 3,100), and (200, 4,200)". Let's use linear regression on these three points. The formula for the y - intercept (b) in linear regression for three points ((x_1,y_1)), ((x_2,y_2)), ((x_3,y_3)) is given by:

[b=\frac{\sum y_i - m\sum x_i}{n}]

where (m=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2})

For (n = 3), (x_1 = 75,y_1 = 2100); (x_2 = 140,y_2 = 3100); (x_3 = 200,y_3 = 4200)

(\sum x_i=75 + 140+200 = 415)

(\sum y_i=2100 + 3100+4200 = 9400)

(\sum x_iy_i=75\times2100+140\times3100 + 200\times4200=157500+434000 + 840000=1431500)

(\sum x_i^2=75^2+140^2+200^2 = 5625+19600 + 40000=65225)

Now, calculate (m):

[m=\frac{3\times1431500-415\times9400}{3\times65225-(415)^2}=\frac{4294500 - 3901000}{195675 - 172225}=\frac{393500}{23450}\approx16.78]

Now, calculate (b):

[b=\frac{9400-16.78\times415}{3}=\frac{9400 - 6963.7}{3}=\frac{2436.3}{3}=812.1). No, that's not right. Wait, no, the formula for linear regression for (n) points is (b=\bar{y}-m\bar{x}), where (\bar{x}=\frac{\sum x_i}{n}), (\bar{y}=\frac{\sum y_i}{n})

(\bar{x}=\frac{415}{3}\approx138.33), (\bar{y}=\frac{9400}{3}\approx3133.33)

[b = 3133.33-16.78\times138.33\approx3133.33 - 2321.23=812.1). Still not matching. Wait, maybe the problem expects us to use two points. Let's use (75,2100) and (140,3100) to find the slope: (m=\frac{3100 - 2100}{140 - 7