Chi-squared goodness of fit testsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A die is thrown 120 times. Faces 1 to 6 occurred 14, 25, 17, 22, 24 and 18 times respectively. A $\chi^2$ goodness of fit test is to be used to test whether the die is fair.

   (a)

   Find the expected frequency for each face if the die is fair.

   [1 mark]

   • A$\frac16$
   • B$6$
   • C$20$
   • D$120$

   (b)

   State the number of degrees of freedom for the test.

   [1 mark]

   • A$5$
   • B$6$
   • C$4$
   • D$119$

   (c)

   Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A firm records the number of machine breakdowns in each of 100 weeks. No breakdowns occurred in 27 weeks, 1 in 35 weeks, 2 in 22 weeks, 3 in 11 weeks, 4 in 4 weeks and 5 in 1 week. The mean number of breakdowns per week is 1.33. The firm models the weekly number of breakdowns by a Poisson distribution with mean 1.33. The expected frequencies are 26.45, 35.18, 23.39, 10.37, 3.45 and 1.17 for 0, 1, 2, 3, 4 and 5 or more breakdowns respectively.

   (a)

   The test requires every expected frequency to be at least 5. Which cells must be combined?

   [1 mark]

   • A4 and 5 or more only
   • B3, 4 and 5 or more
   • C3 and 5 or more only
   • DNo cells need combining, because every observed frequency is at least 1

   (b)

   After combining cells, the firm tests whether the Poisson model fits. State the number of degrees of freedom.

   [1 mark]

   • A$3$
   • B$4$
   • C$1$
   • D$2$

   (c)

   Calculate the value of the test statistic for the combined cells.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An inspector tests 120 boxes, each containing 4 bulbs, and records the number of defective bulbs in each box. There were 48 boxes with no defective bulbs, 42 with 1, 22 with 2, 7 with 3 and 1 with 4. The inspector suggests that the number of defective bulbs in a box can be modelled by $B(4,p)$, with $p$ estimated from the data.

   (a)

   State suitable hypotheses and show that the estimate of $p$ is 0.231 to 3 significant figures.

   [3 marks]

   (b)

   With $p=0.23125$, the expected frequencies for 0, 1, 2, 3 and 4 defective bulbs are 41.91, 50.43, 22.75, 4.56 and 0.34. Carry out the test at the 5% significance level, giving the test statistic and the critical value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A botanist measures the heights of 150 plants. The sample mean is 40 cm and the sample standard deviation is 4 cm. The heights are grouped into five classes: under 34 cm, 34 to 38 cm, 38 to 42 cm, 42 to 46 cm and over 46 cm, containing 14, 31, 55, 38 and 12 plants respectively. The botanist suggests that the heights are modelled by a Normal distribution, with mean and variance estimated from the sample.

   (a)

   (i) Show that the expected frequency for the class 38 to 42 cm is 57.4 to 3 significant figures.
   (ii) Hence find the expected frequencies for the other four classes and calculate the test statistic.

   [6 marks]

   (b)

   The test statistic is 2.92. Carry out a goodness of fit test at the 5% significance level, explaining why no classes need to be combined and how the degrees of freedom are found.

   [6 marks]

   Total for question 4: 12 marks

End of questions