Problem Solving with Potato Weights in Jacob's Crisp Factory

1. GSCE Mathematics Problem Solving Handling Data Higher Tier

2. Helping hand Jacob says that the smallest potato may weigh 100g. Is he correct? Explain your answer. Think about your inequalities. What does 100 < w ≤ 200 mean? Jacob manages a crisp factory. He wants to decide whether or not to use Lady Rosetta, Hermes or Saturna potatoes to make his crisps. The three varieties of potato all cost the same amount per kilogram. Here is some information about a typical bag of each of these varieties of potato. a) This table shows the distribution of weights of potatoes in a bag of 50 Lady Rosetta potatoes.

3. Answer Aron says that the smallest potato may weigh 100g. Is he correct? Explain your answer. 100 < w ≤ 200 Consider the left hand side: 100 < w Is the same as w > 100 Which means that the weight is strictly greater than 100, not greater than or equal to 100, so this is not correct, the potato cannot weigh 100g.

4. Helping hand Frequency = frequency density x class width This histogram shows the distribution of weights (in grams), of potatoes in a bag of Hermes potatoes. How many potatoes were in the bag?

5. Answer This histogram shows the distribution of weights (in grams), of potatoes in a bag of Hermes potatoes. How many potatoes were in the bag? 100 x 0.14 = 14 100 x 0.22 = 22 100 x 0.36 = 36 200 x 0.14 = 28 14 + 22 + 36 + 28 = 100

6. Helping hand You have already worked out the frequency for 300 < w ≤ 500. You need to work this out for 280 < w ≤ 300. This histogram shows the distribution of weights of potatoes in a bag of Hermes potatoes. Find the probability of selecting a potato which is at least 280 grams.

7. This histogram shows the distribution of weights of potatoes in a bag of Hermes potatoes. Find the probability of selecting a potato which is at least 280 grams. Answer You need the area to the right of this line 28 20 x 0.36 = 7.2 280 7.2 + 28 = 35.2, so the probability of this is 35.2/100 = 0.352, as there were 100 potatoes in the bag.

8. Jacob needs a variety of potato with a high average weight. He knows that large potatoes are best for making crisps. Use the information provided above to decide which of the three varieties of potato he should buy. Helping hand Use the information provided in the table above, together with the information you have already gathered. You may wish to discuss mean/median/mode/range/ probability of large potatoes This table shows some information from a bag of 25 Saturna potatoes.

9. Answer Jacob needs a variety of potato with a high average weight. He knows that large potatoes are best for making crisps. Use the information provided above to decide which of the three varieties of potato he should buy. If we consider the median value for each potato: Lady Rosetta: 50 potatoes An estimate of the median is the 25th value. This lies in the range 200 < w ≤ 300 Saturna: Values 1-15 The next 23 values i.e. 16- 38 The 25th value lies here

10. Hermes: 100 potatoes. An estimate of the median is the 50th value. The 50th value lies in here, so 200 < w ≤ 300 The next 22 values 15-36 The next 36 values 37-72 14 values 1-14 The median’s for the Lady Rosetta and Hermes potatoes lie between 200 < w ≤ 300. The Saturna potato has the highest median at 320g, i.e. 50% of Saturna potatoes weigh at least 320g. This would be the best potato to choose based on the median.

11. If we also consider the probability of choosing a potato which weighs at least 300g, we can see that: Lady Rosetta: 8 + 4 = 12 potatoes weigh at least 300g 12/50 = 24% Hermes: 28 potatoes weigh at least 300g 28/100 = 28% Saturna: At least 50% of the potatoes weigh more than 320g (from the Median). 28 These results also show that there is a higher proportion of heavier potatoes with the Saturna variety. Jacob should buy Saturna potatoes.