Everyone who participates in a
three-kilometer fun run is awarded a medal. The table shows the medals awarded
last year. Part a) Estimate the mean time
taken to complete the race last year.
There’s also a part b, but we’ll
come to there in a bit. So in this question, what we’re
looking to do is actually estimate the mean. The reason we say “estimate” is
actually because we don’t know the exact times taken by the people who participate
because it says that we have, for instance, seven people who actually participate
and get a time between six and 20 minutes and get a gold medal. But we don’t know the exact
times. We just know that between six and
20 minutes so that’s why whatever we do is gonna be an estimate.
Now in order to actually estimate
the mean from a frequency table, what we actually have is this formula. In this formula is that the mean —
in this case 𝑥 bar — is equal to the sum of 𝑓𝑥 over the sum of 𝑓, where the
Greek symbol Σ actually means sum so that’s why we have Σ 𝑓𝑥 means the sum of 𝑓
multiplied by 𝑥 divided by the sum of 𝑓.
But what does this mean in actual
practice? Well, what it means is actually the
sum of everything that actually occurs because it’s our frequencies multiplied by
our times then divided by the actual sum of the number of different outcomes or I
assume in this case different people who have participated.
Well, in order to use this formula,
what we need to do is actually know what our 𝑓 and 𝑥 are going to be. Well, first of all, the 𝑓 is
pretty straightforward cause our 𝑓 is just gonna be our frequency and I’ve marked
this in pink above the column. Now, in order to actually find our
𝑥, so what’s the 𝑥 going to be in our formula, well, this is actually going to be
And what we mean by the midpoint is
actually a time in this case between the two values that we have in our class
interval. And the reason we need a midpoint
is cause as I mentioned earlier we don’t know the actual times of the individual
participants. So therefore, what we need to do is
actually decide what we’re gonna take for that value. And what we take is the midpoint,
hence why it’s an estimate of the mean.
Now, in order to actually work out
what the midpoint is for each class interval, what we do is we actually add the two
values in our interval together, so six and 20, and then we divide it by two. So we’d have 26 divided by two,
which is 13. Now, we do this to actually work
out the value of our midpoint for each of the sections. And when we do that, we have
midpoints of 13 as we already mentioned, 27, 41, and 55.
So now, what we do is we actually
had another column. And this column is called 𝑓𝑥. And if we look back at our formula,
we need an 𝑓𝑥. And what this is going to be is our
frequency multiplied by our midpoint. So our first value in this column
is actually going to be 91. And that’s because that’s seven
multiplied by 13. And then if we continue down the
column, we’re gonna get 2025 because that’s 75 times 27, 1517 — 37 multiplied by
41. And then, finally, if you multiply
six and 55, we get 330. So that’s it. We’ve completed the two columns
that we’ve added.
So now, the next stage, if we have
a look, is actually to work out what the sum of 𝑓 and the sum of 𝑓𝑥 are because
they’re what we need to actually fulfill our formula. Well, the sum of 𝑓 what that means
is actually if we add together all our frequencies. And when we actually calculate
this, what we’re gonna get is 125. So therefore, we can say that
actually 125 people participated in the three-kilometer fun run.
So then, what we need to do is
actually work out well what’s gonna be the sum of 𝑓𝑥. Well, sum of the 𝑓𝑥 is all the
𝑓𝑥 values added together. So we get 91 plus 2025 plus 1517
plus 330, which gives us total 𝑓𝑥 sum of 3963. So now, what we can actually do is
work out our estimate for our mean. So we can say that our mean or our
𝑥 bar is the actual notation we’re using is equal to 3963 because this was the sum
of 𝑓𝑥 divided by 125 because this was the sum of 𝑓, which gives us an answer of
31.704. Okay, let’s actually round this to
a sensible degree of accuracy. So therefore, if we round it to one
decimal place, which seems pretty sensible with this answer, we’re gonna get 31.7 as
our estimate for our mean.
Okay, so we found the answer. So that’s great. What I always say is actually check
to see if it actually makes sense. So looking back at the table, we
can see that actually, yeah, the value looks sensible because we got 31.7, which
fits into the time period between 20 and 34. So yes, we know it’s a proper
value. That makes sense. If it was a 0.2 or 400, we’d know
that actually that wouldn’t fit the values in the table. And also we can see that the
frequency is 75. So actually, a large number of our
participants were actually in this class. So the mean makes sense.
Okay, let’s move on to part b of
the question. So for part b, we actually get a
bit more information.
Ben is using last year’s results to
decide how many medals to order this year. He will order exactly as many
medals as participants. Since the modal time is in the
silver medal category, half of the medals that he orders will be silver. Edgar says that this might not be
enough. Who is correct? Give a reason to justify your
Well, the first thing to look at is
actually this, which is the modal time. So what does the modal time
actually mean? Well, the modal actually means the
most common, so the most common time. Well, therefore, if we actually
look just to double check that he is correct and it is in silver medal category, we
can look at where is the most frequency, so where are the most participants.
Well, if we take a look at the
table, we can see yes, most participants are actually in the silver medal, so the
position. And that’s because we have a
frequency of 75, whereas the others we have a frequency of seven, 37, and six. And actually 75 is the greatest out
of these. So yes, this is definitely where
the modal time would lie.
But now, what we need to see is
actually will ordering half of the medals as silver medals mean that there are
actually gonna be enough. Well, if we look at last year’s
data in the table, what we can say is that actually 75 out of 125 of the medals were
actually silver because what we can say is that 75 silver medals were awarded over
125 because there are 125 medals awarded in total.
So now, what we want to do is
actually divide both the numerator and denominator by 25. And we do this so that we can
actually simplify our fraction. And if we do this, we’re gonna get
three-fifths. And that’s because 75 divided by 25
is three and 125 divided by 25 is five. So therefore, actually based on
last year’s medals, we can see that last year actually three-fifths of the medals
were going to be silver and that’s actually greater than a half.
So therefore, we can say that in
answer to the question, who is correct? Edgar is correct. And that’s because more than half
the participants received a silver medal last year because three-fifths of the
participants received silver medal and three-fifths is greater than a half.