EXM.1.AHL.TZ0.59

pestleMathematicsAIHLPaper 1EXM· sl-4-9-normal-distribution-and-calculationssource ↗

The heights, $x$ metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.

$\sum {x = 412.11,\,\,\sum {{x^2} = 705.5721} }$

The Head of Mathematics decided to use a ${\chi ^2}$ test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.

Calculate unbiased estimates of the population mean and the population variance.

[3]

State suitable hypotheses.

[1]

b.i.

Calculate the value of the ${\chi ^2}$ statistic and state your conclusion using a 10% level of significance.

[11]

b.ii.

Markscheme / solution

$\bar x = \frac{{412.11}}{{241}} = 1.71$ A1

${s^2} = \frac{{705.5721}}{{240}} - \frac{{{{412.11}^2}}}{{240 \times 241}} = 0.0036$ M1A1

[3 marks]

H₀: Data can be modelled by a normal distribution

H₁: Data cannot be modelled by a normal distribution A1

[1 mark]

b.i.

The expected frequencies are

A1A1A1A1A1A1

${\chi ^2} = \frac{{{5^2}}}{{8.04}} + \frac{{{{34}^2}}}{{30.19}} + \ldots + \frac{{{{12}^2}}}{{16.10}} - 241 = 3.30\,{\text{/}}\,3.29$ M1A1

Degrees of freedom = 3 A1

Critical value = 6.251 or p-value = 0.35 A1

The data can be modelled by a normal distribution. R1

[11 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.