OCR Further Statistics (Further Statistics) 2021 June

A set of bivariate data \((X, Y)\) is summarised as follows. \(n = 25\), \(\Sigma x = 9.975\), \(\Sigma y = 11.175\), \(\Sigma x^2 = 5.725\), \(\Sigma y^2 = 46.200\), \(\Sigma xy = 11.575\)

1. Calculate the value of Pearson's product-moment correlation coefficient. [1]
2. Calculate the equation of the regression line of \(y\) on \(x\). [2]

It is desired to know whether the regression line of \(y\) on \(x\) will provide a reliable estimate of \(y\) when \(x = 0.75\).

1. State one reason for believing that the estimate will be reliable. [1]
2. State what further information is needed in order to determine whether the estimate is reliable. [1]

The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520. [3]
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval. [1]

The greatest weight \(W\) N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.

1. Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
2. State an assumption needed in carrying out the test in part (a). [1]
3. Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]

The random variable \(D\) has the distribution Geo\((p)\). It is given that Var\((D) = \frac{40}{9}\). Determine

1. Var\((3D + 5)\). [1]
2. E\((3D + 5)\). [6]
3. \(\text{P}(D > \text{E}(D))\). [3]

A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726. Test at the 5% significance level whether there is a difference in the ranks of the students taught by the two professors. [10]