Question

the table shows data from a party planner, representing the number of people at an event (x) and the total dollar cost to host the event (y).
x | 50 | 75 | 100 | 125 | 140 | 150 | 175 | 200 | 220
y | 1,550 | 2,100 | 2,425 | 2,900 | 3,100 | 3,500 | 3,800 | 4,200 | 4,400
when using the median - fit method with summary points (75, 2,100), (140, 3,100), and (200, 4,200), what is the approximate y - intercept of the best - fit model? round to the nearest whole number if necessary.
○ 748
○ 750
○ 809
○ 840

Explanation:

Step1: Find the slope between two points

First, we find the slope \( m \) between two pairs of points. Let's take \((75, 2100)\) and \((140, 3100)\). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). So, \( m=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.3846 \). Let's check with another pair, say \((140, 3100)\) and \((200, 4200)\). \( m=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.3333 \). Wait, maybe we should use the median - fit method's approach. The median - fit line is a linear model \( y = mx + b \), where \( b \) is the y - intercept. We can use two points to find the slope first. Let's use \((75,2100)\) and \((200,4200)\). Then \( m=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8 \).

Step2: Use the slope to find the y - intercept

Now that we have the slope \( m = 16.8 \), we can use the point - slope form \( y - y_1=m(x - x_1) \). Let's use the point \((75,2100)\). So, \( y-2100 = 16.8(x - 75) \). Expand the right - hand side: \( y-2100=16.8x-1260 \). Then, solve for \( y \): \( y = 16.8x-1260 + 2100 \), \( y = 16.8x + 840 \)? Wait, no, that's not right. Wait, let's recalculate the slope correctly. Let's use two points properly. Let's take \((75,2100)\) and \((140,3100)\): \( m=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.3846 \). Then use the point \((75,2100)\) in \( y=mx + b \). So, \( 2100=15.3846\times75 + b \). Calculate \( 15.3846\times75 = 15.3846\times70+15.3846\times5 = 1076.922+76.923 = 1153.845 \). Then \( b=2100 - 1153.845 = 946.155 \). No, that's not matching. Wait, maybe I made a mistake. Let's use the three points. The median - fit line is calculated by dividing the data into three groups. But we are given the summary points \((75,2100)\), \((140,3100)\), and \((200,4200)\). The median of the x - values (75,140,200) is 140, and the median of the y - values (2100,3100,4200) is 3100. But for the linear model \( y=mx + b \), we can use two points to find the slope. Let's use \((75,2100)\) and \((200,4200)\). Slope \( m=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8 \). Now, substitute \( x = 75 \), \( y = 2100 \) into \( y=mx + b \): \( 2100=16.8\times75 + b \). Calculate \( 16.8\times75=1260 \). Then \( b = 2100 - 1260=840 \)? No, that's not. Wait, wait, maybe the slope is calculated as the average of the slopes between the three points. The slope between \((75,2100)\) and \((140,3100)\) is \( m_1=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.3846 \). The slope between \((140,3100)\) and \((200,4200)\) is \( m_2=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.3333 \). The median of \( m_1\) and \( m_2\) is the average of \( 15.3846\) and \( 18.3333\)? Wait, no, the median - fit slope is the median of the slopes of the lines connecting the first and second, second and third, and first and third points. The slopes are:

• Between (75,2100) and (140,3100): \( m_1=\frac{3100 - 2100}{140 - 75}=\frac{1000}{65}\approx15.38 \)
• Between (140,3100) and (200,4200): \( m_2=\frac{4200 - 3100}{200 - 140}=\frac{1100}{60}\approx18.33 \)
• Between (75,2100) and (200,4200): \( m_3=\frac{4200 - 2100}{200 - 75}=\frac{2100}{125}=16.8 \)

The median of 15.38, 16.8, 18.33 is 16.8. Now, use the point (140,3100) (the median x and median y point) in \( y = mx + b \). So, \( 3100=16.8\times140 + b \). Calculate \( 16.8\times140 = 2352 \). Then \( b=3100 - 2352 = 748 \). Let's check with the point (75,2100): \( y=16.8\times75+748=1260 + 748 = 2008 \), which is close to 2100. With (200,4200): \( y=16.8\times200+748 = 3360+748 = 4108 \), close to 4200. So the y - intercept \(…

Answer:

748