WST01 Statistics 1 (S1): May/June 2021 Paper 01

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.

(a)Complete the tree diagram.

[2]

Given that Gina has selected a yellow counter,

(b)find the probability that she has 2 counters.[3]

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 P(A) = 0.65

(a)State which two of the events A, B and C are mutually exclusive.[1]

(b)Find the value of r and the value of s.[5]

The events (A ∩ C′) and (B ∪ C) are also independent.

(c)Find the exact value of p and the exact value of q. Give your answers as fractions.[6]

A random sample of 100 carrots is taken from a farm and their lengths, L cm, recorded. The data are summarised in the following table.

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

(a)Find the width and height of the bar representing the class 15  L < 20[3]

(b)Use linear interpolation to estimate the median length of these carrots.[2]

(c)Estimate

(i)the mean length of these carrots,[2]

(ii)the standard deviation of the lengths of these carrots.[3]

A supermarket will only buy carrots with length between 9 cm and 22 cm.

(d)Estimate the proportion of carrots from the farm that the supermarket will buy.[2]

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.

(e)Find an estimate of the mean profit per carrot made by the farm.[2]

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.

(a)Estimate the proportion of letters sent by the company that weigh less than 120 g.[3]

Kris splits the letters to be sent into 3 categories: heavy, medium and light, with 13 of the letters in each category.

(b)Find the weight limits that determine medium letters.[4]

A heavy letter is chosen at random.

(c)Find the probability that this letter weighs less than 200 g.[3]

Kris chooses a random sample of 3 letters from those in the mailroom one day.

(d)Find the probability that there is one letter in each of the 3 categories.[3]

The discrete random variable X has the following probability distribution

Given that E(X ) = 0.5

(a)find the value of a.[2]

Given also that Var(X ) = 5.01

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

The random variable Y = 5 − 8X

(c)Find (i) E(Y )

(ii)Var(Y )[3]

(d)Find P(4X² > Y)[5]

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

(a)Show that S_yy = 7.524 375[1]

(b)Find S_xx[4]

(c)Find the product moment correlation coefficient between x and y.[3]

Behrouz claims that, assuming the model is valid, the data show that when unemployment is 2% wages increase at over 3%

(d)Explain how Behrouz could have come to this conclusion.[1]

Andi uses the formula

to estimate the range of values for x.

(e)Find estimates of the minimum value and the maximum value of x in these data using Andi’s formula.[3]

(f)Comment, giving a reason, on the reliability of Behrouz’s claim.[2]

Andi suggests using the regression line with equation y = 3.684 – 0.3242x to estimate unemployment when wages are increasing at 2%

(g)Comment, giving a reason, on Andi’s suggestion.[2]