WST01 Statistics 1 (S1): Oct/Nov 2022 Paper 01

The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

A histogram is drawn to represent these data.

The area of the bar representing the 40  x 42 class is 16.5 cm

(a)Calculate the exact area of the bar representing the 42  x 45 class.[2]

The height of the tallest bar in the histogram is 10 cm.

(b)Find the exact height of the second tallest bar.[3]

Q₁ for these data is 45 cm.

(c)Use linear interpolation to find an estimate for

(i)Q₂

(ii)the interquartile range.[4]

One measure of skewness is given by

(d)By calculating this measure, describe the skewness of these data.[2]

The production cost, £c million, of a film and the total ticket sales, £t million, earned by the film are recorded for a sample of 40 films.

Some summary statistics are given below.

(a)Find the exact value of S_tt and the exact value of S_ct[3]

(b)Calculate the value of the product moment correlation coefficient for these data.[2]

(c)Give an interpretation of your answer to part (b)[1]

(d)Show that the equation of the linear regression line of t on c can be written as

where the values of the intercept and gradient are given to 3 significant figures.[3]

(e)Find the expected total ticket sales for a film with a production cost of £90 million.[2]

Using the regression line in part (d)

(f)find the range of values of the production cost of a film for which the total ticket sales are less than 80% of its production cost.[2]

Morgan is investigating the body length, b centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using

The results for the coded data are summarised below

(a)Find the mean of b[3]

(b)Find the standard deviation of b[3]

A 9th squirrel is added to the sample. Given that for all 9 squirrels x = 0

(c)find

(i)the body length of the 9th squirrel,[2]

(ii)the standard deviation of x for all 9 squirrels.[2]

The cumulative distribution function of the discrete random variable W, which takes only the values 6, 7 and 8, is given by

[4]

The weights, W grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where k is a positive constant.

One kiwi fruit is selected at random from those grown on the farm.

(a)Find the probability that this kiwi fruit is Large.[3]

35% of the kiwi fruit are Jumbo.

(b)Find the value of k to one decimal place.[4]

75% of Tiny kiwi fruit weigh more than y grams.

(c)Find the value of y giving your answer to one decimal place.[5]

The Venn diagram shows the events A, B, C and D, where p, q, r and s are probabilities.

(a)Write down the value of

(i)P(A)

(ii)P(A½B)

(iii)P(A½C )[3]

(b)find the exact value of q and the exact value of r[6]

Given also that P(B ∪ C¢) = 58

(c)find the exact value of s[2]

Adana selects one number at random from the distribution of X which has the following probability distribution.

(a)Given that the number selected by Adana is not 5, write down the probability it is 0[1]

(b)Show that E(X²) = 75[1]

(c)Find Var(X )[3]

(d)Find Var(4 – 3X )[2]

Bruno and Charlie each independently select one number at random from the distribution of X

(e)Find the probability that the number Bruno selects is greater than the number Charlie selects.[3]

Devika multiplies Bruno’s number by Charlie’s number to obtain a product, D

(f)Determine the probability distribution of D[4]