SPS SPS ASFM Statistics (SPS ASFM Statistics) 2021 May

1. The complex number \(3 + 2i\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w^*\). Find
   1. the modulus of \(w\), [1]
   2. the argument of \(w^*\), giving your answer in radians, correct to 2 decimal places. [3]
2. Find the complex number \(u\) given that \(u + 2u^* = 3 + 2i\). [4]
3. Sketch, on an Argand diagram, the locus given by \(|z + 1| = |z|\). [2]

1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]

Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\).

1. Find the point of intersection of \(l_1\) and \(l_2\). [4]
2. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]

The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other. [3]
2. Find the probability that no two men are next to one another. [4]

Every time a spinner is spun, the probability that it shows the number 4 is 0.2, independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required. [2]
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive. [2]
3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\). [4]

Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.

1. Arlosh calculates Spearman's rank correlation coefficient \(r_s\) for the critics' ratings. He calculates that \(\Sigma d^2 = 72\). Show that this value must be incorrect. [2]
2. Arlosh checks his working with Sarah, whose answer \(r_s = \frac{39}{35}\) is correct. Find the correct value of \(\Sigma d^2\). [2]
3. Carry out an appropriate two-tailed significance test of the value of \(r_s\) at the 5% significance level, stating your hypotheses clearly. [4]

A spinner has edges numbered 1, 2, 3, 4 and 5. When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that E\((N) = 2.55\).

1. Find Var\((N)\). [7]
2. Find E\((3N + 2)\). [1]
3. Find Var\((3N + 2)\). [1]

A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.

1. Find the probability of exactly 4 faults in a 15 metre length of cloth. (2)
2. Find the probability of more than 10 faults in 60 metres of cloth. (3)

A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80

1. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. (4)

The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.

1. Find the retailer's expected profit. (4)