A local shop holds a competition where customers are asked to guess the number of sweets in a jar. The table shows information about today’s guesses. The correct number of sweets in the jar is exactly 1.5 fewer sweets than the mean of today’s guesses. How many of today’s guesses were too high? You must show your working.
So in this question, the first thing we need to do is work out the mean number of sweets guessed. So what we’re going to do is find that mean from a frequency table because this is what we have here. We have a frequency table that tells us that 23 sweets were guessed once, 28 sweets was guessed four times, 29 sweets was guessed 11 times, 30 sweets were guessed five times, 32 sweets were guessed 17 times, and 36 sweets were guessed twice. And we know that the mean, is equal to the total number of sweets guessed divided by the total number of guesses. And we can work this out by writing out 23 once, 28 four times, 29 11 times, etcetera, and then adding them all together and dividing them by how many different numbers there were.
But this is long winded. So we can do it much more easily using our frequency table. So what we do is we call the number of sweets guessed 𝑥. And we got the frequency 𝑓. And then we create this extra column called 𝑓𝑥. So this means 𝑓 multiplied by 𝑥 or frequency multiplied by number of sweets guessed. And what we can do at this stage is introduced some new notation. We’ve got 𝑥 bar, which means mean, is equal to 𝛴𝑓𝑥, where 𝛴𝑓𝑥 is the sum of 𝑓𝑥 because 𝛴 means sum, divided by 𝛴𝑓. So now, what we’re gonna do is work out what 𝑓𝑥 is. So we’ve got 23 multiplied by one, which is 23. Then 28 multiplied by four, we can use our calculator, is 112. Then we have 29 multiplied by 11 which is 319, 30 multiplied by five, which is 150, 32 multiplied by 17 which is 544, 36 multiplied by two, which is 72.
So now, what we want to do is find out the total of 𝑓𝑥 if we add them all together. And when we do that, we get 1220. And that’s because that’s 23 add 112 add 319 add 150 add 544 add 72. So now, we can use our formula to find the mean. And that’s because our mean is gonna be equal to 1220 which was our some of 𝑓𝑥 divided by our some of 𝑓, so the sum of the frequencies which is 40. And using our calculator, we get a mean of 30.5. So great, we’ve found the mean. Have we finished the question? Well, no, because we want to work out how many of today’s guesses are too high. Well, we’re told that the correct number of sweets in the jar is exactly 1.5 fewer sweets than the mean. So therefore, the number of sweets in the jar is gonna be equal to 30.5 minus 1.5.
We can do this using a calculator or mentally. If we do it mentally. If we take away 0.5, that takes us to 30, then take away one takes us to 29. So we know that there are 29 sweets in the jar. But still, we haven’t quite finished because we want to work out how many today’s guesses were too high. Well, if we look at the guesses that were 23. These aren’t too high. Well, the guesses that we’re 28, these aren’t too high cause they’re not greater than 29. And then if we look at the next row, it’s 29 is the number of sweets guessed. Well, this is not too high because it says how many of today’s guesses were too high. Well, this is exactly the correct guess. So therefore, this is not the one that we’re looking for, which means, therefore, that the next three rows are all too high. So the guess is in the next three rows.
Be careful at this point because a common mistake would be to say, well, the number of guesses that were too high was three because it’s 30, 32, and 36. But this is not correct because it’s the frequencies that we need to look at. So therefore, the number of guesses too high is gonna be five cause that’s the number of people who guessed 30 sweets add 17, the number of people who guessed 32 sweets, add two, the number of people who guessed 36 sweets. Well five add 17 is 22 add two is 24. So therefore, we can say that the number of guesses that were too high was 24 guesses.