WST01 Statistics 1 (S1): January 2021 Paper 01

The Venn diagram shows the events A, B and C and their associated probabilities.

(a)P(Bʹ)[1]

(b)P(A ∪ C)[2]

(c)P(A | Bʹ )[2]

The stem and leaf diagram below shows the ages (in years) of the residents in a care home.

(a)Find the median age of the residents.[1]

(b)Find the interquartile range (IQR) of the ages of the residents.[2]

An outlier is defined as a value that is either

more than 1.5 ´ (IQR) below the lower quartile or

more than 1.5 ´ (IQR) above the upper quartile.

(c)Determine any outliers in these data. Show clearly any calculations that you use.[3]

(d)On the grid on page 5, draw a box plot to summarise these data.[3]

The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg

(a)Find the probability that a randomly selected package weighs less than 10 kg[3]

The heaviest 15% of packages are moved around the factory by Jemima using a forklift truck.

(b)Find the weight, in kg, of the lightest of these packages that Jemima will move.[3]

One of the packages not moved by Jemima is selected at random.

(c)Find the probability that it weighs more than 18 kg[4]

A delivery of 4 packages is made to the factory. The weights of the packages are independent.

(d)Find the probability that exactly 2 of them will be moved by Jemima.[3]

A spinner can land on the numbers 10, 12, 14 and 16 only and the probability of the spinner landing on each number is the same. The random variable X represents the number that the spinner lands on when it is spun once.

(a)State the name of the probability distribution of X.[1]

(b)(i)Write down the value of E(X )[1]

(ii)Find Var(X )[2]

A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable Y represents the number that this spinner lands on when it is spun once. The probability distribution of Y is given in the table below

(c)Find (i) E(Y )[2]

(ii)Var(Y )[3]

The random variable W = aX + b, where a and b are constants and a > 0 Given that E(W ) = E(Y ) and Var(W ) = Var(Y )

(d)find the value of a and the value of b.[5]

Each of the two spinners is spun once.

(e)Find P(W = Y )[2]

A company director wants to introduce a performance-related pay structure for her managers. A random sample of 15 managers is taken and the annual salary, y in £1000, was recorded for each manager. The director then calculated a performance score, x, for each of these managers. The results are shown on the scatter diagram in Figure 1 on the next page.

(a)Describe the correlation between performance score and annual salary.[1]

The results are also summarised in the following statistics.

(b)(i)Show that S_xy = 2006[1]

(ii)Find S_yy[2]

(c)Find the product moment correlation coefficient between performance score and annual salary.[2]

The director believes that there is a linear relationship between performance score and annual salary.

(d)State, giving a reason, whether or not these data are consistent with the director’s belief.[1]

(e)Calculate the equation of the regression line of y on x, in the form y = a + bx Give the value of a and the value of b to 3 significant figures.[4]

(f)Give an interpretation of the value of b.[1]

(g)Plot your regression line on the scatter diagram in Figure 1[2]

The director hears that one of the managers in the sample seems to be underperforming.

(h)On the scatter diagram, circle the point that best identifies this manager.[1]

The director decides to use this regression line for the new performance related pay structure.

(i)Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30[2]

Turn over for a spare copy of the scatter diagram if you need to redraw your line.

A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.

If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won.

By considering the possible positions for the centre of the disc,

(a)show that the probability of winning a prize on any particular roll is 15[3]

A group of 15 students each roll the disc onto the grid twenty times and record the number of times, x, that each student wins a prize. Their results are summarised as follows

(b)Find the standard deviation of the number of prizes won per student.[2]

A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2

(c)Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students.[7]

The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).

(d)Explain how this probability could be used to find an estimate for the value of π and state the value of your estimate.[3]