EXM.1.SL.TZ0.11

pestleMathematicsAISLPaper 1EXM· sl-4-11-expected-observed-hypotheses-chi-squared-gof-t-testsource ↗

A calculator generates a random sequence of digits. A sample of 200 digits is randomly selected from the first 100 000 digits of the sequence. The following table gives the number of times each digit occurs in this sample.

It is claimed that all digits have the same probability of appearing in the sequence.

Test this claim at the 5% level of significance.

[7]

Explain what is meant by the 5% level of significance.

[2]

Markscheme / solution

H₀: The sequence contains equal numbers of each digit. (A1)

H₁: The sequence does not contain equal numbers of each digit. (A1)

$\chi _{{\text{calc}}}^2 = \frac{{\left( {9 + 1 + 25 + 1 + 25 + 49 + 1 + 9 + 4 + 16} \right)}}{{20}} = 7$ (M1)(A1)

The number of degrees of freedom is 9. (A1)

$\chi _{0.95{\text{;}}\,\,5}^2 = 16.919$ (A1)

$\chi _{{\text{calc}}}^2 < 16.919$ . Hence H₀ is accepted. (A1)

[7 marks]

The probability of rejecting H₀ when it is true (A1)

is 0.05. (A1)

Note: Award (A1)(A1) for “the probability of a type I error is 0.05.”

[2 marks]

Examiners’ report

[N/A]