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CAPE Applied Mathematics Unit 2 · 2016 · Paper 2 · Question 3(b)(iii)

Data on stem density x (stems/m²) and stem width y (cm) for 10 plants: ∑x = 130, ∑y = 5.5, ∑xy = 59.95, ∑x² = 2090, ∑y² = 3.875.

The correlation coefficient is r = -0.63. Interpret this value in the context of the data.

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  1. 3(a)A random sample of 49 items with a sample mean of 13 is taken from a normal distribution with standard deviation 4. Calculate a 96% confidence interval for the…[3 marks]
  2. 3(b)(i)Determine the estimated linear regression equation y = a + bx for this data.[5 marks]
  3. 3(b)(ii)Estimate the width of a stem when the stem density is 8 using your regression equation.[2 marks]
  4. 3(c)(i)State appropriate null and alternative hypotheses to test for association between gender and rating of restroom facilities using a chi-squared test.[2 marks]
  5. 3(c)(ii)Determine the critical region for the chi-squared test at the 5% significance level.[2 marks]
  6. 3(c)(iii)Calculate the expected frequency corresponding to cell (row 2, column 2), which has an observed value of 24.[2 marks]
  7. 3(c)(iv)a)State whether you reject or fail to reject the null hypothesis.[1 mark]
  8. 3(c)(iv)b)Interpret the test decision in the context of the problem.[1 mark]
  9. 3(a)(i)Calculate the total number of distinct portfolios of 12 paintings that can be created.[3 marks]
  10. 3(a)(ii)Determine the number of portfolios that contain exactly 8 watercolours and 4 oil paintings.[4 marks]
  11. 3(a)(iii)Hence, calculate the probability that a randomly chosen portfolio contains 8 watercolours and 4 oil paintings.[2 marks]
  12. 3(b)(i)Calculate the probability that a player starts the game on the third toss of the die.[4 marks]
  13. 3(b)(ii)Calculate the probability that at most 5 tosses are necessary to start the game.[3 marks]
  14. 3(b)(iii)Determine the expected number of tosses required to start the game.[2 marks]
  15. 3(c)(i)Calculate the probability that exactly 4 faulty reports are received on Monday.[3 marks]
  16. 3(c)(ii)Calculate the probability that exactly 5 faulty reports are received over a five-day period.[4 marks]

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