Lesson video

Hello, my name's Miss Parnham, and in this lesson, we're going to learn how to design and interpret two-way tables.

Here, we have an example of a two-way table that has been partially completed.

It's a two-way table because we have two pieces of information here, the sandwich choice and the fruit choice for a school packed lunch.

So we can see that the columns are the sandwich fillings and the rows are the fruit choices.

And what we know so far is 24 people chose a chicken sandwich with an apple.

And the numbers in this table are going to support us finding all the missing values, and that's what we'll do now.

So probably the first one we can complete is if we know the total is 100 and 56 people chose banana, then everybody else chose apple.

So that is going to be 44, 100 subtract 56 gets us that value.

And we nearly have a complete row for those that picked apple, so we know that the chicken and the egg sandwich eaters sums to 33 apples, and it must be an extra 11 to make 44 in total.

So subtracting 24 and nine from 44 gets us 11, and from this, we can total up the number of cheese sandwich choices.

11 and 12 add together to make 23.

And we can see that the egg sandwich choices must add to 24, and if nine of those chose apple then 15 must have chosen banana because nine and 15 sum to 24.

Now, it actually doesn't matter which way around we find these pieces of information.

We could subtract 15 and 12 from 56 to find out how many people chose a chicken sandwich and a banana, and that's 29.

And then to find the total, we can sum 24 and 29 to get 53.

And just as a double check, we can add 53, 24 and 23 and find that that does indeed make 100.

Right, we have a question about probability.

So a person is chosen at random, so it's out of the total number of people, which is 100, what is the probability they chose an egg sandwich and a banana? So look in the column for egg and the row for banana, meet at 15, So that's 15/100, or any percentage or decimal or equivalent fraction to that is fine.

A person has chosen an apple, so we've put in a condition on this now.

So it's only out of the people who chose apple, so that's 44, we worked that out.

What is the probability they are having a cheese sandwich? So that is 11/44, you can simplify that to 1/4 if you want, and we know that that's equivalent to 0.

25 or 25%, any of those are fine as a solution to that question.

Here's a question for you to try, pause the video to complete the task and restart the video when you're finished.

Here are the answers, in part B, you are asked for a ratio, and the order is very important.

So we're asked for the ratio of children to adults, so we should have 38 to 82.

And if you've written that, give yourself some credit, but that can be simplified further because we have a common factor of two, so 19 to 41 is as simple as it goes.

Here's another question for you to try, pause the video to complete the task, and then restart the video when you're finished.

Here are the answers, did you spot the total in the introductory text? That is going to be key to help you find that 99 people opted for chicken.

And when we're asked for the probability that someone picked at random chose chicken, then 99/180 is completely acceptable.

It does not have to be simplified, but if you did simplify it, any fraction equivalent to that, and the simplest is 11/20, or if you gave it as a percentage, 55%, or is a decimal, 0.

55, they are both acceptable as answers when asked for probability.

In this example, we have a lot of information and we have to design the two-way table to accommodate it.

So we always must read the full text first to appreciate how the data is being categorised.

And from this, we can see that students study one of three languages and they either do that part time or full time.

So it doesn't matter whether you choose to make the type of language the column or the row.

But if we lay this out like this, we've made the, French, Spanish and German are our columns, and part time and full time run across our rows.

Right, the first piece of information in this text tells us that 200 students are at this language school.

So that is the ground total and goes there in the bottom right hand corner.

And next we're told that 136 students study part time, so that's the total of all the part time students, is 136, of which 61 study French.

So where the French column meets the part time row, we can input 61.

Now, there's a piece of information here that we can add, because we know the total students is 200 and there are 136 doing the courses part time, we can subtract 136 from 200 and workout that 64 are full time.

The next line of information tells us that 74 students study Spanish, that's in total, so we can place that there, and 31 of these are full time.

So that's where Spanish meets full time.

So what we can deduce from this is that 74 subtract 31 gives us 43 part time Spanish students.

Of those people studying German, the ratio of part time to full time is four to one.

Well, we don't know any of that information yet, but we can glean it from what we already inputted into the table.

So 61 and 43 make a total of 104, and if we subtract that from 136, this gives us 32 German students studying part time.

So if part time to full time is four to one, then that's the same as 32 to eight, multiply those by eight.

So yes, they will total 40, place that in the table, and then to work out full time French students and total French students, We can do that either way.

We can either subtract 31 and eight from 64, and that will get us 25, and then we can add 61 and 25 together to get total French students as 86.

And 86, 74 and 40 do make 200, so that is all correct.

What's the probability that a student chosen at random is studying French full time? So that's out of all the students, so it's over 200, and the full time French is 25, so we can write 25/200, or 12.

5% 0.

125, or any equivalent fraction to that.

What is the probability that a full time student is studying French? Oh, that's phrased slightly differently, full time students total 64.

And we know that 25 are studying French, so that's 25/64.

Here's a question for you to try, pause the video to complete the task and then restart the video when you're finished.

Here are the answers, from the frequency tree, we can see that the total is 80 and we can work out that there must be 24 people who chose paddle boarding, and of these, six must have then picked rock climbing.

Then we have 27 of the mountain bikers going sailing in the afternoon, and all of these numbers will then translate to the two-way table, but you'll notice that the two-way table gives us slightly more information than the frequency tree because we can see the totals for the rock climbing and sailing.

Here's a further question for you to try, pause the video to complete the task and then restart the video when you're finished.

Here are the answers, can you imagine trying to solve this problem without a two-way table? It helps us organise the data and find a logical path through until we've found all the missing values.

So in part B, we need to be very careful of the wording because it says a car and not a vehicle, so we're picking a car at random, so it is only from the 76 cars.

And 59 of these are not silver, so that's where that probability originates from.

That's all for this lesson, thank you for watching.