The weights of 60 students are shown. Part a) Circle the class interval which contains the median. 50 is less than or equal to 𝑤 which is less than 60, 60 is less than or equal to 𝑤 which is less than 70, 70 is less than or equal to 𝑤 which is less than 80, or 80 is less than or equal to 𝑤 which is less than 90.
The data we’ve been given is organised into a frequency table. It shows that our data — in this case the weight of the students — ranges from a minimum of 40 up to but not equal to 100 kilograms. In each row, we have a range of weights and then the frequency of students that occur within that range.
We can imagine this data on a number line. Our first row is a weight greater than or equal to 40 but less than 50 and three students are in this range. We don’t know exactly the weight of these three students, but we do know that they fall in this range. The next class up greater than or equal to 50 but less than 60 has 16 students, greater than or equal to 60 but less than 70 has 20 students.
Part a wants to know which of these four intervals the median falls into. Here are our four choices for the median. We remember that the median is the middle value in an ordered list of numbers. Our weights are ordered from least to greatest in these classes. To find the middle value, we need to take in the number of data points we have add one and then divide by two. That would be 60 plus one divided by two which equals 30.5. And that means in an ordered list, the median is going to fall between the 30th value and the 31st value.
To find the 30th and 31st value, we can use the cumulative frequency. The cumulative frequency of the first row is still just going to be three. The cumulative frequency of the first class interval is the same as its original frequency. But to find the cumulative frequency of the second row, we add the frequencies from the first two rows: three plus 16 equals 19. This cumulative frequency of 19 tells us that there were 19 students whose weight was greater than 40 but less than 60.
For the next row, we need to add up the first three classes: three plus 16 plus 20 equals 39. That means that 39 of the students weighed more than 40 kilograms but less than 70. Remember we’re looking for the 30th value and the 31st value.
If 19 values fall between greater than or equal to 40 and less than 60 and 39 values are greater than or equal to 40 but less than 70, that means inside this class — this third class — are all the values from 20 to 39. And that means the 30th value and the 31st value would fall in this class.
Because class three has a cumulative frequency of 39 and the previous class had a cumulative frequency of 19, we know that the 30th value and the 31st value fall in this class greater than or equal to 60 but less than 70. And that means the median will fall here. The median will fall in the weight class of 𝑤 is greater than or equal to 60 but less than 70.
Part b) A teacher says, “40 percent of the students weigh at least 70 kilograms.” Does the information support the teacher’s claim? You must give reasons for your answer.
The teacher is claiming that 40 percent of the students weigh at least 70 kilograms. First, let’s consider how many students it would take to equal 40 percent, 40 percent of the students. When we see the word “of,” we know that we’re dealing with multiplication. We need to multiply 40 percent times the total number of students which is 60. The fraction form of 40 percent is 40 out of 100. 40 over 100 times 60.
We can rewrite the numerator to say four times six times 100 and the denominator is still 100. Those hundreds cancel out. And we see that four times six equals 24. 40 percent of the 60 students would be 24 students. This is the first part of her claim.
The second part is that they weigh at least 70 kilograms. Which of these classes contain students that weigh at least 70 kilograms? The fourth class begins at 70 kilograms. All the students in the fifth class have to weigh more than 80 kilograms but less than 90. And all the students in the sixth class need to weigh more than 90 kilograms but less than 100.
On our number line, we can see that these three classes are the classes that contain values that are equal to or greater than 70. The first one has 12 students, the second one has seven students, and the third one has two students. When we add those values up, we see that 21 students weigh at least 70 kilograms. We know that 21 students weighed at least 70 kilograms.
We can say that in order for 40 percent of the students to weigh at least 70 kilograms, 24 students would need to be 70 or more kilograms. Since only 21 students weigh at least 70 kilograms, the claim is not true. Don’t forget that this question is asking for the reasons to support your answer and so all of your working is important here.