WST01 Statistics 1 (S1): January 2023 Paper 01

The histogram shows the times taken, t minutes, by each of 100 people to swim 500 metres.

(a)Use the histogram to complete the frequency table for the times taken by the 100 people to swim 500 metres.

[1]

(b)Estimate the number of people who took less than 16 minutes to swim 500 metres.[2]

(c)Find an estimate for the mean time taken to swim 500 metres.[2]

Given that ft² = 41 033 ∑

(d)find an estimate for the standard deviation of the times taken to swim 500 metres.[2]

Given that Q₃ = 23

(e)use linear interpolation to estimate the interquartile range of the times taken to swim 500 metres.[3]

Two bags, X and Y, each contain green marbles (G) and blue marbles (B) only.

• Bag X contains 5 green marbles and 4 blue marbles

• Bag Y contains 6 green marbles and 5 blue marbles

A marble is selected at random from bag X and placed in bag Y A second marble is selected at random from bag X and placed in bag Y A third marble is then selected, this time from bag Y

(a)Use this information to complete the tree diagram shown on page 7[3]

(b)Find the probability that the 2 marbles selected from bag X are of different colours.[2]

(c)Find the probability that all 3 marbles selected are the same colour.[2]

Given that all three marbles selected are the same colour,

(d)find the probability that they are all green.[3]

9

The probability distribution of the discrete random variable X is given by

where a is a constant.

(a)Find, in terms of a, E(X)[2]

(b)Find the range of the possible values of E(X)[3]

Given that Var(X) = 0.56

(c)find the possible values of a[6]

(i)In the Venn diagram below, A and B represent events and p, q, r and s are probabilities.

(a)Use algebra to show that 2p + 2q + 2r = 45[4]

(b)Find the value of p, the value of q, the value of r and the value of s[5]

(ii)Two events, C and D, are such that

where x is a positive constant.

By considering P(C) + P(D) show that C and D cannot be mutually exclusive.[4]

The lengths, L mm, of housefly wings are normally distributed with L ~ N(4.5, 0.4²)

(a)Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm.[3]

(b)Find

(i)the upper quartile (Q₃) of L

(ii)the lower quartile (Q₁) of L[4]

A value that is greater than Q₃+ 1.5 × (Q₃– Q₁) or smaller than Q₁– 1.5 × (Q₃– Q₁) is defined as an outlier.

(c)Find these two outlier limits.[3]

A housefly is selected at random.

(d)Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places.[3]

Given that this housefly is not an outlier,

(e)showing your working, find the probability that the wing length of this housefly is greater than 5 mm.[4]

A research student is investigating the maximum weight, y grams, of sugar that will dissolve in 100 grams of water at various temperatures, x °C, where 10  x  80

The research student calculated the regression line of y on x and found it to be

(a)Give an interpretation of the gradient of the regression line.[1]

(b)Use the regression line to estimate the maximum weight of sugar that will dissolve in 100 grams of water when the temperature is 90 °C.[2]

(c)Comment on the reliability of your estimate, giving a reason for your answer.[2]

Using the regression line of y on x and the following summary statistics

(d)show that the product moment correlation coefficient for these data is 0.988 to 3 decimal places.[7]

The research student’s supervisor plotted the original data on a scatter diagram, shown on page 23

With reference to both the scatter diagram and the correlation coefficient,

(e)discuss the suitability of a linear regression model to describe the relationship between x and y.[2]