How to solve Problem solving and Data Analysis questions in SAT Math
The Problem Solving and Data Analysis section in SAT Math tests the student’s ability to use their skills related to quantitative reasoning about ratios, rates, and proportional relationships. You are expected to identify quantitative measures of center, overall patterns, and any unusual data from the overall pattern and spread in one or two different data sets. This includes recognizing the effects of outliers on the measures of center of a data set.
SAT Math Problem solving and Data Analysis Sample problem solved
Questions 1-3 refer to the following information.
The first metacarpal bone is located in the wrist. The scatterplot below shows the relationship between the length of the first metacarpal bone and height for 9 people. The line of best fit is also shown.
- How many of the nine people have an actual height that differs by more than 3 centimeters from the height predicted by the line of best fit?
Select an Answer
Lets look at the sentences above the graph. It talks about a certain bone (metacarpal) and how the graph shows the relationship between the height of a person and his length of the first metacarpal bone.
We see here a bunch of dots (or points). Also, a straight dark line. What do the dots mean?
They indicate the actual data collected from the nine men. e.g. See the first dot, which is on the y axis. The coordinates of the point is (4, 157). i.e. The height of this man is 157 cms and the length of his first metacarpal bone is 4 cm.
Similarly, the other dots represent the data for the other 8 men.
What does the line represent?
Since the data is scattered, the easiest way is to draw a line that can approximately depict the relationship between the height of the men and the length of their bone. Why the line is drawn as it is shown, is a different topic of discussion, which we wont dwell here. However, the line helps us now to predict the height of any person if their length of the first metacarpal bone is known. For example if your first metacarpal bone is say 5cm long, then looking at the graph and the line, it can be predicted that your height is 180 cm.
So our question is about the number of people whose actual height differs by more than 3 cms from the height predicted from the line.
Look at the graph. Each horizontal line is separated by a distance of 1 cm. So for us to find out the answer to our problem, we need to find how many of the dots are below or above the line by 3 cms. By observation, if you go from left to right of the graph, dots 1, 3, 7 and 8 are above or below by at least 3 horizontal lines i.e. by 3 cms or more.
Therefore, the correction answer is option B) 4.
Hope you understood the step by step method and if practiced, you should be able to solve this or similar problem in less than 2 minutes.
Now, lets go to the second question of this problem.
2) Which of the following is the best interpretation of the slope of the line of best fit in the context of this problem?
A) The predicted height increase in centimeters for one centimeter increase in the first metacarpal bone
B) The predicted first metacarpal bone increase in centimeters for every centimeter increase in height
C) The predicted height in centimeters of a person with a first metacarpal bone length of 0 centimeters
D) The predicted first metacarpal bone length in centimeters for a person with a height of 0 centimeters
the question is basically asking what the slope of the line in the graph represents.
Look at the line. It is slanting upwards from left to right. This means it has a positive slope. i.e. As we move to the right on the x axis, the value of y increases.
The x axis in the graph starts at 4 cm and ends at 5 cm. Therefore, the graph shows the relationship for the length of the first metacarpal bone increase of 1 cm.
Options C and D are eliminated because, there is nothing on the graph which represents 0 cm length of metacarpal bone or 0 cm height of a person.
Option B is incorrect, since the slope is the change in vertical distance with change in horizontal distance.
Option A is correct since the slope shows how the y axis (height of the person) changes with the x axis (length of the first metacarpal bone).
Lets look at the 3rd question of the problem.
3) Based on the line of best fit, what is the predicted height for someone with a first metacarpal bone that has a length of 4.45 centimeters?
This question is testing your skill of understanding the graph.
It is asking the height corresponding the metacarpal bone length of 4.45 cms.
Look at the x axis. The distance between 4.0 and 4.5 (and between 4.5 and 5.0) is separated by 5 Vertical line.
Thus each vertical line separation is (4.5- 4.0)/5 = 0.1 cm
So 4.4 is the fouth vertical line after 4.0. Since 4.45 cm is the average of 4.4 and 4.5 cm, the x coordinate is between the 4th vertical line and line representing 4.5 cm. Using your scale, you see that the point on the line at x = 4.45 cms, corresponds to 170 cms on the y=axis.
Hence the answer is option C). 170 cms.
Hope this helped you how to approach such type of problems.
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