describing the median - fit method\na restaurant manager records the number of tables seated (x) and the…

Explanation:

Step1: Arrange data and find summary - points

First, arrange the data in ascending order of (x) - values. For the median - fit method, divide the data into three equal - sized groups. The left - most group gives the left summary point, the middle group gives the middle summary point, and the right - most group gives the right summary point. The left - most group: ((35,99.5),(38,97.25),(41,102)), the median of (x) in this group is (38) and the median of (y) is (97.25), so the left summary point is ((38,97.25)). The middle group: ((47,110.5),(52,103.75)), the median of (x) is (\frac{47 + 52}{2}=49.5), the median of (y) is (\frac{110.5+103.75}{2}=107.125) (it seems there is a mistake in the problem - statement for the middle summary point). The right - most group: ((55,125),(58,160),(63,142),(71,153)), the median of (x) is (\frac{58 + 63}{2}=60.5), the median of (y) is (\frac{160 + 142}{2}=151) (it seems there is a mistake in the problem - statement for the right summary point).

Step2: Calculate slope and y - intercept (if correct summary points are known)

The formula for the slope (m) of the line (y=mx + b) using the three summary points ((x_1,y_1)), ((x_2,y_2)), ((x_3,y_3)) is (m=\frac{(y_3 - y_1)(x_2 - x_1)-(y_2 - y_1)(x_3 - x_1)}{(x_3 - x_1)(x_2 - x_1)}). And the (y) - intercept (b) can be found by substituting one of the summary points into (y=mx + b) and solving for (b). But with the correct summary points ((38,97.25)), if we assume the correct middle point and right point and calculate: Let's assume the correct process. The left summary point ((x_1,y_1)=(38,97.25)), assume the middle point ((x_2,y_2)=(52,110.5)) (as per the problem - statement for checking), and assume the right point ((x_3,y_3)). The slope (m=\frac{(y_3 - y_1)(x_2 - x_1)-(y_2 - y_1)(x_3 - x_1)}{(x_3 - x_1)(x_2 - x_1)}). If we use the two - point formula for slope between ((38,97.25)) and ((52,110.5)): (m=\frac{110.5 - 97.25}{52 - 38}=\frac{13.25}{14}\approx0.946\neq2.14). To find the (y) - intercept, using (y=mx + b) with ((x = 38,y = 97.25)) and (m\approx0.946), we have (97.25=0.946\times38 + b), (b=97.25-0.946\times38=97.25 - 35.948 = 61.302\neq - 10.48).

Answer:

The left summary point is ((38,97.25))