Question 1

A merchant has found that the number of items of two brands A and B that she can sell in a day are independent Poisson random variables with means 4 and 10 respectively.

(a) What is the probability that no item of brand A will be sold in a day? What is the probability that no item of either brand will be sold in a day?

(6 marks)

(b) If the merchant has 8 items of brand A and 6 items of brand B in stock, what are the probabilities that she will sell out her stock of each brand individually and of both brands together in one day?

(6 marks)

(c) What is the probability distribution and the expected number of days out of 25 that the merchant will sell no items of brand A?

(6 marks)

(d) What is the probability of selling out all the stock in (b) in two days?

(7 marks)

Question 2

A  report  on  the  latest A-level  results  achieved  by  students  from  a  particular  school showed that the average A-level points (per student) had dropped from 41 in 2018 to 39.5 in 2019. A journalist from a local newspaper had written a feature on this claiming that the points drop resulted from a fall in teaching standards. The school had played this down, claiming that it was simply due to random fluctuations from year to year. The standard deviation of points achieved by the 100 students who took A-levels in 2018 was 8.2, whereas in 2019 the 112 students who took A-levels had a standard deviation of 9.0.

(a) Carry out an appropriate test to see whether there has been a significant drop in A-level scores in the school from 2018 to 2019. Justify your choice of test, stating clearly any assumptions that you have made. Use a  5% significance level and state your conclusion clearly.

(10 marks)

(b) While  the  journalist’s  article  had  criticised  overall  standards  at  the  school, individual departments were keen to demonstrate that they were not to blame. The History department was particularly adamant that their exam results had actually improved in 2019. The head of the department recalled that in 2018, 23 students specialised in History, achieving on average 39.2 A-level points, with a standard deviation of 8.6. In 2019, 29 students specialised in History, achieving an average of 44.6 points, with a standard deviation of 9.6.

Carry  out  an  appropriate  statistical test to  investigate this  possibility  for the History department. Justify your choice of test, stating clearly any assumptions that  you  have  made.  Use  a  5%  significance  level  and  state  your  conclusion clearly.

(15 marks)

Question 3

(a) Lancaster metal products sells its product in three primary colours: red, blue and yellow. The marketing department believes that customers have no colour preference for the product. To see if this is true the product manager performs a test in which 120 purchasers were given equal opportunity to buy the product in each of three colours. The results were that 60 bought red, 20 bought blue, and 40 bought yellow. Perform a

hypothesis test to check the marketing department’s belief at a 5% level. (10 marks)

(b) The Analytics teaching department wanted to understand the student absences and recorded the absences to their courses in last 106 days, which is summarised as below:

The department believes the data follows Normal distribution. The sample mean and standard deviation of the 106 days of data are 141.68 and 5.56 respectively. Using this

information test the department’s hypothesis, at 1% level. (15 marks)

Question 4

NO2 concentration is believed to be a major contributor to temperature and overall         pollution. Data has been collected on roads of London through several sensors to collect several air quality measures including temperature in degrees Celsius and NO2                  concentration in mg/m3. To understand the relationship between the variables,                 temperature has been regressed against NO2, and the results are shown in the following pages.

(a) Comment on how well the regression line provides a good fit to the data.

(5 marks)

(b) Is there evidence of a significant relationship? (5 marks)

(c) Assess the assumptions of the regression model using all the information given

below. (10 marks)

(d) Comment on the predictions output by SPSS for NO2 concentrations 5mg/m3

and 150mg/m3 . (5 marks)