Video Transcript
The data table shows the high jump
and long jump results achieved by 15 competitors in the women’s heptathlon in the
2016 Rio Olympics. Calculate, to the nearest thousand,
the value of the product-moment correlation coefficient between the long jump and
high jump results. What does this correlation
coefficient reveal about the relationship between the long jump and high jump
results?
We’re given a table of values for
two variables, long jump and high jump scores for 15 women athletes in the Rio
Olympics. This is bivariate data, which means
that two measurements are recorded for each individual athlete, how far they jumped
in the long jump and how high they jumped in the high jump. So, for example, athlete one jumped
5.51 meters in the long jump and 1.65 meters in the high jump. And there are two parts to this
question. We’re first asked to calculate the
product-moment correlation coefficient and then we’re asked for an interpretation of
this value.
For the first part of the question,
we’re going to use the formula shown for the correlation coefficient 𝑟 subscript
𝑥𝑦 which you may see written as 𝑟. And to use this formula, we recall
that the capital Σ symbol signifies the sum, also that 𝑛 is the number of data
points or pairs. In our case, we have 15 athletes so
that 𝑛 is equal to 15. So let’s begin by calling the long
jump in meters the variable 𝑋 and the high jump in meters the variable 𝑌. To calculate our coefficient, we’re
going to need the various expressions within the formula. We’ll need the products 𝑥𝑦, the
𝑥-values squared, and the 𝑦-values squared. And so, we add some rows to our
table to help us with our calculations. We also add a column to the end of
our table for our sums.
So let’s first work out the
products 𝑥𝑦 for each athlete. For our first athlete, the product
is 5.51 multiplied by 1.65. And that’s equal to 9.0915. And so we put this into the first
empty cell of our new row for 𝑥𝑦. Similarly, for our second athlete,
we have 5.72 multiplied by 1.77, which is 10.1244. And this goes into the second cell
for the 𝑥𝑦 row. And so we continue in this way,
filling in the rest of the row, where we’ve restricted ourselves to three decimal
places for the sake of space. In the second new row of our table,
we want the 𝑥-values squared. So for example, our first entry
will be 5.51 squared, and that’s 30.3601. Putting this in our table, we
restrict ourselves now to two decimal places for the sake of space. And squaring the remainder of our
𝑥-values, we can fill in the table as shown.
Next, we take the squares of the
high jump values, that’s the 𝑦-values squared, and fill in our table as shown. So now let’s fill in our sums
column, where the sum of the 𝑋’s, for example, is the long jump scores all
summed. And that’s equal to 91.43. The sum of all high jump scores,
that’s the sum of the 𝑌-values, is 27.21. The sum of the products 𝑥𝑦 is
166.1151; that’s to four decimal places. The sum of the 𝑥-values squared is
558.4923. And the sum of the squares of the
𝑦’s is 49.4361. So in our sums column, we have the
sum of the 𝑋-values, the sum of the 𝑌-values, the sum of the product 𝑥𝑦, the sum
of the 𝑥-values squared, and the sum of the 𝑦-values squared. So now, we have everything we need
for our formula.
With 𝑛 is 15, that’s the number of
athletes, and all of the sums from our table, our correlation coefficient can be
calculated as shown. Using our calculators, we can
evaluate the numerator and denominator as shown. So we have 3.9162 divided by
4.5567, each to four decimal places, which is 0.8594 to four decimal places. To three decimal places then, that
is, to the nearest thousandth, the value of the product-moment correlation
coefficient between the long jump and high jump results is 0.859.
For the second part of the question
regarding the relationship between the long jump and high jump results, our
coefficient is very close to positive one. This means that there’s a strong
positive, that is, direct, linear correlation between long jump and high jump
results for the women athletes in the Rio Olympics.
