Edexcel S1 (Statistics 1) 2021 June

1. There are 7 red counters, 3 blue counters and 2 yellow counters in a bag. Gina selects a counter at random from the bag and keeps it. If the counter is yellow she does not select any more counters. If the counter is not yellow she randomly selects a second counter from the bag.
   1. Complete the tree diagram.
   First Counter
   Second Counter
   \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-02_1147_1081_603_397} Given that Gina has selected a yellow counter,
2. find the probability that she has 2 counters.

2. In the Venn diagram below, \(A , B\) and \(C\) are events and \(p , q , r\) and \(s\) are probabilities. The events \(A\) and \(C\) are independent and \(\mathrm { P } ( A ) = 0.65\)
\includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}

1. State which two of the events \(A\), \(B\) and \(C\) are mutually exclusive.
2. Find the value of \(r\) and the value of \(s\). The events ( \(A \cap C ^ { \prime }\) ) and ( \(B \cup C\) ) are also independent.
3. Find the exact value of \(p\) and the exact value of \(q\). Give your answers as fractions.

1. A random sample of 100 carrots is taken from a farm and their lengths, \(L \mathrm {~cm}\), recorded. The data are summarised in the following table.

A histogram is drawn to represent these data.
The bar representing the class \(5 \leqslant L < 8\) is 1.5 cm wide and 1 cm high.

1. Find the width and height of the bar representing the class \(15 \leqslant L < 20\)
2. Use linear interpolation to estimate the median length of these carrots.
3. Estimate
   1. the mean length of these carrots,
   2. the standard deviation of the lengths of these carrots. A supermarket will only buy carrots with length between 9 cm and 22 cm .
4. Estimate the proportion of carrots from the farm that the supermarket will buy. Any carrots that the supermarket does not buy are sold as animal feed. The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
5. Find an estimate of the mean profit per carrot made by the farm.

1. Kris works in the mailroom of a large company and is responsible for all the letters sent by the company. The weights of letters sent by the company, \(W\) grams, have a normal distribution with mean 165 g and standard deviation 35 g .
   1. Estimate the proportion of letters sent by the company that weigh less than 120 g .
   Kris splits the letters to be sent into 3 categories: heavy, medium and light, with \(\frac { 1 } { 3 }\) of the letters in each category.
2. Find the weight limits that determine medium letters. A heavy letter is chosen at random.
3. Find the probability that this letter weighs less than 200 g . Kris chooses a random sample of 3 letters from those in the mailroom one day.
4. Find the probability that there is one letter in each of the 3 categories.

1. The discrete random variable \(X\) has the following probability distribution

Given that \(\mathrm { E } ( X ) = 0.5\)

1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)

1. Two economics students, Andi and Behrouz, are studying some data relating to unemployment, \(x \%\), and increase in wages, \(y \%\), for a European country. The least squares regression line of \(y\) on \(x\) has equation

$$y = 3.684 - 0.3242 x$$ and $$\sum y = 23.7 \quad \sum y ^ { 2 } = 42.63 \quad \sum x ^ { 2 } = 756.81 \quad n = 16$$

1. Show that \(\mathrm { S } _ { y y } = 7.524375\)
2. Find \(\mathrm { S } _ { x x }\)
3. Find the product moment correlation coefficient between \(x\) and \(y\). Behrouz claims that, assuming the model is valid, the data show that when unemployment is 2\% wages increase at over 3\%
4. Explain how Behrouz could have come to this conclusion. Andi uses the formula $$\text { range } = \text { mean } \pm 3 \times \text { standard deviation }$$ to estimate the range of values for \(x\).
5. Find estimates of the minimum value and the maximum value of \(x\) in these data using Andi's formula.
6. Comment, giving a reason, on the reliability of Behrouz's claim. Andi suggests using the regression line with equation \(y = 3.684 - 0.3242 x\) to estimate unemployment when wages are increasing at \(2 \%\)
7. Comment, giving a reason, on Andi's suggestion.
   
   \includegraphics[max width=\textwidth, alt={}]{a439724e-b570-434d-bf75-de2b50915042-20_2647_1835_118_116}