Edexcel S1 (Statistics 1)

1. Draw two separate scatter diagrams, each with eight points, to illustrate the relationship between \(x\) and \(y\) in the cases where they have a product moment correlation coefficient equal to
   1. exactly \(+1\),
   2. about \(-0.4\). [4 marks]
2. Explain briefly how the conclusion you would draw from a product moment correlation coefficient of \(+0.3\) would vary according to the number of pairs of data used in its calculation. [2 marks]

A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course. There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,

1. find the height of the rectangle representing this group. [3 marks]

Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm,

1. find the number of people on the course in this age group. [3 marks]

The events \(A\) and \(B\) are such that $$\text{P}(A) = \frac{7}{12}, \quad \text{P}(A \cap B) = \frac{1}{4} \quad \text{and} \quad \text{P}(A|B) = \frac{2}{3}.$$ Find

1. P\((B)\), [3 marks]
2. P\((A \cup B)\), [3 marks]
3. P\((B|A')\). [3 marks]

The owner of a mobile burger-bar believes that hot weather reduces his sales. To investigate the effect on his business he collected data on his daily sales, \(£P\), and the maximum temperature, \(T\)°C, on each of 20 days. He then coded the data, using \(x = T - 20\) and \(y = P - 300\), and calculated the summary statistics given below. $$\Sigma x = 57, \quad \Sigma y = 2222, \quad \Sigma x^2 = 401, \quad \Sigma y^2 = 305576, \quad \Sigma xy = 3871.$$

1. Find an equation of the regression line of \(P\) on \(T\). [9 marks]

The owner of the bar doesn't believe it is profitable for him to run the bar if he takes less than £460 in a day.

1. According to your regression line at what maximum daily temperature, to the nearest degree Celsius, does it become unprofitable for him to run the bar? [3 marks]

The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Find the value of \(k\). [2 marks]
2. Show that E\((X) = \frac{9}{2}\). [3 marks]

Find

1. P\([X > \text{E}(X)]\), [2 marks]
2. E\((2X - 5)\), [2 marks]
3. Var\((X)\). [4 marks]

A geologist is analysing the size of quartz crystals in a sample of granite. She estimates that the longest diameter of 75% of the crystals is greater than 2 mm, but only 10% of the crystals have a longest diameter of more than 6 mm. The geologist believes that the distribution of the longest diameters of the quartz crystals can be modelled by a normal distribution.

1. Find the mean and variance of this normal distribution. [9 marks]

The geologist also estimated that only 2% of the longest diameters were smaller than 1 mm.

1. Calculate the corresponding percentage that would be predicted by a normal distribution with the parameters you calculated in part \((a)\). [3 marks]
2. Hence, comment on the suitability of the normal distribution as a model in this situation. [2 marks]

Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.

1. Construct a stem and leaf diagram for these data. [3 marks]
2. Find the median and quartiles for these data. [4 marks]
3. Represent these data with a boxplot. [3 marks]

Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.

1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]