Edexcel S1 (Statistics 1) 2023 June

1. The histogram shows the distances, in km , that 274 people travel to work.
   \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-02_1272_1582_296_175}

Given that 60 of these people travel between 10 km and 20 km to work, estimate

1. the number of people who travel between 22 km and 45 km to work,
2. the median distance travelled to work by these 274 people,
3. the mean distance travelled to work by these 274 people.

1. Two students, Olive and Shan, collect data on the weight, \(w\) grams, and the tail length, \(t \mathrm {~cm}\), of 15 mice.

Olive summarised the data as follows $$\mathrm { S } _ { t t } = 5.3173 \quad \sum w ^ { 2 } = 6089.12 \quad \sum t w = 2304.53 \quad \sum w = 297.8 \quad \sum t = 114.8$$

1. Calculate the value of \(\mathrm { S } _ { t w }\) and the value of \(\mathrm { S } _ { w w }\)
2. Calculate the value of the product moment correlation coefficient between \(w\) and \(t\)
3. Show that the equation of the regression line of \(w\) on \(t\) can be written as $$w = - 16.7 + 4.77 t$$
4. Give an interpretation of the gradient of the regression line.
5. Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2 cm . Shan decided to code the data using \(x = t - 6\) and \(y = \frac { w } { 2 } - 5\)
6. Write down the value of the product moment correlation coefficient between \(x\) and \(y\)
7. Write down an equation of the regression line of \(y\) on \(x\) You do not need to simplify your equation.

1. Jim records the length, \(l \mathrm {~mm}\), of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained.

$$n = 81 \quad \sum x = 3711 \quad \sum x ^ { 2 } = 475181$$

1. Find the mean length of these salmon.
2. Find the variance of the lengths of these salmon. The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below.
   \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-10_362_1479_849_296}
3. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ Raj says that the box plot is incorrect as Jim has not included outliers.
   For these data an outlier is defined as a value that is more than
   \(1.5 \times\) IQR above the upper quartile or \(1.5 \times\) IQR below the lower quartile
4. Show that there are no outliers.

1. A bag contains a large number of coloured counters. Each counter is labelled A, B or C
   \(30 \%\) of the counters are labelled A
   \(45 \%\) of the counters are labelled B
   The rest of the counters are labelled C
   It is known that
   2\% of the counters labelled A are red
   4\% of the counters labelled B are red
   6\% of the counters labelled C are red
   One counter is selected at random from the bag.
   1. Complete the tree diagram on the opposite page to illustrate this information.
   2. Calculate the probability that the counter is labelled A and is not red.
   3. Calculate the probability that the counter is red.
   4. Given that the counter is red, find the probability that it is labelled C
   \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}

6\% of the counters labelled C are red
One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information.
2. Calculate the probability that the counter is labelled A and is not red.
3. Calculate the probability that the counter is red.
4. Given that the counter is red, find the probability that it is labelled C \end{enumerate} \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
   5. A discrete random variable \(Y\) has probability function $$\mathrm { P } ( \mathrm { Y } = \mathrm { y } ) = \left\{ \begin{array} { c l } \mathrm { k } ( 3 - \mathrm { y } ) & y = 1,2
   \mathrm { k } \left( \mathrm { y } ^ { 2 } - 8 \right) & y = 3,4,5
   \mathrm { k } & y = 6
   0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
5. Show that \(k = \frac { 1 } { 30 }\) Find the exact value of
6. \(\mathrm { P } ( 1 < Y \leqslant 4 )\)
7. \(\mathrm { E } ( Y )\) The random variable \(X = 15 - 2 Y\)
8. Calculate \(\mathrm { P } ( Y \geqslant X )\)
9. Calculate \(\operatorname { Var } ( X )\)
   1. Three events \(A , B\) and \(C\) are such that
   $$\mathrm { P } ( A ) = 0.1 \quad \mathrm { P } ( B \mid A ) = 0.3 \quad \mathrm { P } ( A \cup B ) = 0.25 \quad \mathrm { P } ( C ) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
10. find \(\mathrm { P } ( A \cup C )\)
11. Show that \(\mathrm { P } ( B ) = 0.18\) Given also that \(B\) and \(C\) are independent,
12. draw a Venn diagram to represent the events \(A , B\) and \(C\) and the probabilities associated with each region.

1. A machine squeezes apples to extract their juice. The volume of juice, \(J \mathrm { ml }\), extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\)

Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to

1. 1. show that \(\mathrm { P } ( J > 510 ) = 0.3446\)
   2. calculate the value of \(d\) such that \(\mathrm { P } ( J > d ) = 0.9192\) Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
2. Calculate the probability that each of the 5 bags of apples produce less than 510 ml of juice. Following adjustments to the machine, the volume of juice, \(R \mathrm { ml }\), extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that \(\mathrm { P } ( R < r ) = 0.15\) and \(\mathrm { P } ( R > 3 r - 800 ) = 0.005\)
3. find the value of \(r\) and the value of \(k\)