Edexcel Jan 2020 IAL Pure Maths WST01/01

Edexcel Jan 2020 IAL Pure Maths WST01/01 (pdf)

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1. The discrete random variable X has the following probability distribution where a, b and c are probabilities.
   The mean value of X is 1 and F(1) = 0.63
   Find the value of a, the value of b and the value of c.
2. A group of 40 families was asked whether their family had a dog, a cat or a rabbit as pets.
   No family had a dog and a rabbit
   2 families had both a dog and a cat
   12 families had a dog
   14 families had a cat
   11 families had a rabbit
   9 families did not have any of these animals as pets
   A family from this group is selected at random.
   D represents the event that the family has a dog
   C represents the event that the family has a cat
   R represents the event that the family has a rabbit
   (a)Draw a Venn diagram to illustrate this information.
   (b)State, giving a reason, a pair of mutually exclusive events from D, C and R.
   (c)Find the probability that the family has exactly 2 of these kinds of animals as pets.
   (d)Showing your working clearly, determine whether or not the events D and C are independent.
   Sarah’s family is in the group and her family has a pet cat.
   (e)Find the probability that Sarah’s family also has a pet rabbit.
   (f)Find the exact value of P([D∪R]|Cʹ)

3. Soapern sells washing machines. When a customer buys a washing machine from Soapern, the customer is also invited to buy a guarantee policy to cover breakdowns and repairs for the next three years.
   The manager of Soapern believes that the relationship between the number of washing machines sold (m) and the number of guarantee policies sold (p) can be modelled by a straight line.
   She collected data each month for 10 months. The scatter diagram below illustrates these data. The data are summarised by the following statistics.
   (a)Show that S_mp = 1373.6
   (b)Find the value of the product moment correlation coefficient for these data.
   (c)State, giving a reason, whether or not the data are consistent with the manager’s belief.
   The manager noticed that the total number of washing machines sold was k times the total number of guarantee policies sold and suggests a model of the form p=1/k m, where k is an integer.
   (d)Find the value of k.
   Jiang works for Soapern and thought that this model oversimplified the situation and suggested that a linear regression of p on m may be more appropriate.
   (e)Find the equation of the linear regression of p on m, giving your answer in the form p = a + bm, where a and b should be given to 3 significant figures.
   (f)Use Jiang’s model to estimate the number of guarantee policies sold when 70 washing machines are sold in a month.
   Usually about 70 washing machines are sold in January. Soapern decides to offer a bonus to staff during January based on the number of guarantee policies sold. If the number of guarantee policies sold is greater than the number estimated by the model, the bonus will be paid.
   (g)State, giving your reasons, whether you would recommend that the staff use the manager’s model or Jiang’s model.
   
4. A researcher is studying the birth weights of babies. A random sample of 98 babies was taken and their birth weights, wkg, are summarised in the table below.
   A histogram is drawn to represent these data.
   The bar representing the birth weight 1.50 ≤ w < 2.50 has a width of 1cm and a height of 4cm.
   (a)Calculate the width and height of the bar representing birth weight 3.50 ≤ w < 4.00
   (b)Use linear interpolation to estimate the lower quartile of the birth weights of the 98 babies.
   The researcher estimated the median to be 3.14kg and the upper quartile to be 3.55kg.
   (c)Use the median and quartiles to describe the skewness of these data.
   (d)Find an estimate for
   (i)the mean birth weight
   (ii)the standard deviation of the birth weights.
   (e)Use the formula
   to estimate a value for the skewness of these data. Give your answer to 2 significant figures.
   The researcher read that birth weights should be approximately normally distributed and decides to split the class 3.00 ≤ w < 3.50
   The frequency for 3.00 ≤ w < 3.25 is 9 and the frequency for 3.25 ≤ w < 3.50 is 23
   (f)(i)State, giving a reason, what the effect would be on the estimate of the median.
   (ii)Without carrying out any further calculations state, giving a reason, what the effect of this change would be on the estimate of the mean.
   
5. The random variable X has a normal distribution with mean 10 and standard deviation 6 (a)Find P(X < 7)
   (b)Find the value of k such that
   P(10 – k < X < 10 + k) = 0.60
   A single observation x, of X, is to be taken.
   A rectangle is drawn on a centimetre grid with vertices having coordinates (0, 0), (x, 0), (x, x – 3) and (0, x – 3)
   (c)Find the probability that the area of this rectangle is more than 40cm²
6. A tennis tournament has 5 rounds. After each round, winners go into the next round and losers are knocked out of the tournament. To enter the tournament players must pay an entry fee of $10 but only the person who wins all 5 rounds receives the prize of $260.
   Serena enters this tennis tournament. The random variable S represents the total number of rounds Serena wins. The probability distribution for S is given in the following table.
   (a)Show that k=20/49
   (b)Find E(S)
   (c)Find Serena’s expected profit if she enters the tennis tournament.
   Roger also enters this tennis tournament. Given that Roger is still in the tournament, the probability that he wins the next round is a constant p.
   The random variable R represents the total number of rounds that Roger wins.
   (d)Explain why P(R = 2) = p²(1 – p)
   (e)Find, in terms of p, the probability distribution for R.(3)(f)Find the smallest value of p such that Roger’s expected profit is at least as great as Serena’s.

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