Questions AS Paper 2 (315 questions)

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AQA AS Paper 2 Specimen Q17
6 marks Moderate -0.8
The table below is an extract from the Large Data Set.
MakeRegionEngine sizeMassCO2CO
VAUXHALLSouth West139811631180.463
VOLKSWAGENLondon99910551060.407
VAUXHALLSouth West12481225850.141
BMWSouth West297916351940.139
TOYOTASouth West199516501230.274
BMWSouth West297902440.447
FORDSouth West159601650.518
TOYOTASouth West12991050144
VAUXHALLLondon139813611400.695
FORDNorth West495117992990.621
    1. Calculate the standard deviation of the engine sizes in the table. [1 mark]
    2. The mean of the engine sizes is 2084 Any value more than 2 standard deviations from the mean can be identified as an outlier. Using this definition of an outlier, show that the sample of engine sizes has exactly one outlier. Fully justify your answer. [3 marks]
  2. Rajan calculates the mean of the masses of the cars in this extract and states that it is 1094 kg. Use your knowledge of the Large Data Set to suggest what error Rajan is likely to have made in his calculation. [1 mark]
  3. Rajan claims there is an error in the data recorded in the table for one of the Toyotas from the South West, because there is no value for its carbon monoxide emissions. Use your knowledge of the Large Data Set to comment on Rajan's claim. [1 mark]
AQA AS Paper 2 Specimen Q18
4 marks Easy -1.8
Neesha wants to open an Indian restaurant in her town. Her cousin, Ranji, has an Indian restaurant in a neighbouring town. To help Neesha plan her menu, she wants to investigate the dishes chosen by a sample of Ranji's customers. Ranji has a list of the 750 customers who dined at his restaurant during the past month and the dish that each customer chose. Describe how Neesha could obtain a simple random sample of size 50 from Ranji's customers. [4 marks]
AQA AS Paper 2 Specimen Q19
11 marks Standard +0.3
Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution B(\(n\), 0.20).
  1. There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
    1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
    2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
  2. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
    1. Name the sampling method used by Declan. [1 mark]
    2. Describe one weakness of this sampling method. [1 mark]
    3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]
OCR MEI AS Paper 2 2018 June Q1
2 marks Easy -2.0
Write down the value of (A) \(\log_a (a^4)\), [1] (B) \(\log_a \left(\frac{1}{a}\right)\). [1]
OCR MEI AS Paper 2 2018 June Q2
3 marks Easy -1.3
Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2.
Time in minutes\(15-\)\(20-\)\(25-\)\(35-\)\(45-60\)
Number of runners12235971
Frequency density2.45.97.11.4
Fig. 2
  1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet. [2]
  2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version. [1]
OCR MEI AS Paper 2 2018 June Q3
3 marks Moderate -0.8
\(P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\). Prove that \(P^2 - Q^2\) is a multiple of 8. [3]
OCR MEI AS Paper 2 2018 June Q4
5 marks Moderate -0.8
The probability distribution of the discrete random variable \(X\) is given in Fig. 4.
\(r\)01234
P\((X = r)\)0.20.150.3\(k\)0.25
Fig. 4
  1. Find the value of \(k\). [2]
\(X_1\) and \(X_2\) are two independent values of \(X\).
  1. Find P\((X_1 + X_2 = 6)\). [3]
OCR MEI AS Paper 2 2018 June Q5
3 marks Moderate -0.8
Find the set of values of \(a\) for which the equation $$ax^2 + 8x + 2 = 0$$ has no real roots. [3]
OCR MEI AS Paper 2 2018 June Q6
4 marks Moderate -0.8
Show that \(\int_0^9 (3 + 4\sqrt{x})dx = 99\). [4]
OCR MEI AS Paper 2 2018 June Q7
8 marks Moderate -0.8
Rose and Emma each wear a device that records the number of steps they take in a day. All the results for a 7-day period are given in Fig. 7.
Day1234567
Rose10014112621014993619708992110369
Emma9204991387411001510261739110856
Fig. 7 The 7-day mean is the mean number of steps taken in the last 7 days. The 7-day mean for Rose is 10112.
  1. Calculate the 7-day mean for Emma. [1]
At the end of day 8 a new 7-day mean is calculated by including the number of steps taken on day 8 and omitting the number of steps taken on day 1. On day 8 Rose takes 10259 steps.
  1. Determine the number of steps Emma must take on day 8 so that her 7-day mean at the end of day 8 is the same as for Rose. [4]
In fact, over a long period of time, the mean of the number of steps per day that Emma takes is 10341 and the standard deviation is 948.
  1. Determine whether the number of steps Emma needs to take on day 8 so that her 7-day mean is the same as that for Rose in part (ii) is unusually high. [3]
OCR MEI AS Paper 2 2018 June Q8
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]
OCR MEI AS Paper 2 2018 June Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. Research showed that in May 2017 62% of adults over 65 years of age in the UK used a certain online social media platform. Later in 2017 it was believed that this proportion had increased. In December 2017 a random sample of 59 adults over 65 years of age in the UK was collected. It was found that 46 of the 59 adults used this online social media platform. Use a suitable hypothesis test to determine whether there is evidence at the 1% level to suggest that the proportion of adults over 65 in the UK who used this online social media platform had increased from May 2017 to December 2017. [7]
OCR MEI AS Paper 2 2018 June Q10
9 marks Moderate -0.8
  1. A curve has equation \(y = 16x + \frac{1}{x}\). Find
    1. \(\frac{dy}{dx}\), [2]
    2. \(\frac{d^2y}{dx^2}\). [2]
  2. Hence
OCR MEI AS Paper 2 2018 June Q11
9 marks Easy -1.8
The pre-release material contains data concerning the death rate per thousand people and the birth rate per thousand people in all the countries of the world. The diagram in Fig. 11.1 was generated using a spreadsheet and summarises the birth rates for all the countries in Africa. \includegraphics{figure_11_1} Fig. 11.1
  1. Identify two respects in which the presentation of the data is incorrect. [2]
Fig. 11.2 shows a scatter diagram of death rate, \(y\), against birth rate, \(x\), for a sample of 55 countries, all of which are in Africa. A line of best fit has also been drawn. \includegraphics{figure_11_2} Fig. 11.2 The equation of the line of best fit is \(y = 0.15x + 4.72\).
    1. What does the diagram suggest about the relationship between death rate and birth rate? [1]
    2. The birth rate in Togo is recorded as 34.13 per thousand, but the data on death rate has been lost. Use the equation of the line of best fit to estimate the death rate in Togo. [1]
    3. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a country where the birth rate is 5.5 per thousand. [1]
    4. Explain why it would not be sensible to use the equation of the line of best fit to estimate the death rate in a Caribbean country where the birth rate is known. [1]
    5. Explain why it is unlikely that the sample is random. [1]
Including Togo there were 56 items available for selection.
  1. Describe how a sample of size 14 from this data could be generated for further analysis using systematic sampling. [2]
OCR MEI AS Paper 2 2018 June Q12
10 marks Moderate -0.8
In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).
  1. Write down the values of \(A\) and \(k\). [2]
  2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
  3. What does the model suggest about the number of fruit flies in the long run? [1]
Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$
  1. Identify three ways in which this second model is consistent with the known data. [3]
    1. Identify one feature which is not accounted for by the second model. [1]
    2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]