Date | May Example question | Marks available | 8 | Reference code | EXM.1.AHL.TZ0.55 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 0 |
Command term | Test and Calculate | Question number | 55 | Adapted from | N/A |
Question
Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data.
Calculate the mean number of brown eggs in a box.
Hence estimate p, the probability that a randomly chosen egg is brown.
By calculating an appropriate χ2 statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.
Markscheme
Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.
Mean =1×29+…+6×1100=1.98 (A1)
[1 mark]
Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.
ˆp=1.986=0.33 (A1)
[1 mark]
Note: Candidates may obtain slightly different numerical answers depending on the calculator and approach used. Use discretion in marking.
The calculated values are
f0 fe (f0−fe)2
10 9.046 0.910
29 26.732 5.14 (M1)
31 32.917 3.675 (A1)
18 21.617 13.083 (A1)
12 9.688 5.345 (A1)
Note: Award (M1) for the attempt to calculate expected values, (A1) for correct expected values, (A1) for correct (f0−fe)2 values, (A1) for combining cells.
χ2=0.9109.046+…+5.3459.688=1.56 (A1)
OR
χ2=1.56 (G5)
Degrees of freedom = 3; Critical value = 7.815
(or p-value = 0.668 (or 0.669)) (A1)(A1)
We conclude that the binomial distribution does provide a good fit. (R1)
[8 marks]