Statistical inference — Question 14

QCAA 2024, Paper 2 · 5 marks

Q14 · 2024 · Technology-activeSimple familiar5 marks

QUESTION 14 (5 marks)

The height of Year 12 students at a school is normally distributed, with a mean height of 168.6 cm and standard deviation of 12.7 cm.

The heights of a random sample of 20 of these students are recorded.

Explain why it can be assumed that the sample means for random samples of the heights of students from this school are normally distributed.

[1 mark]

Determine the probability that the mean height of this sample will be greater than 170 cm.

[2 marks]

There is a 75% probability that the mean height of this sample will lie within

\pm h

cm of the population mean. Determine

P(\bar X\ge168.6+h)

[1 mark]

Use your result from Question 14c) to determine the value of

h

[1 mark]

WORKED SOLUTION

5 marks

ANSWER

P(\bar X>170)\approx0.31

and

h\approx3.27

cm.

Worked solution

a) The distribution of sample means is normally distributed because the population from which a random sample is taken is normally distributed.

\mu_{\bar X}=168.6\text{ cm},\qquad \sigma_{\bar X}=\frac{12.7}{\sqrt{20}}\approx2.84\text{ cm}.

Using a GDC,

P(\bar X>170)\approx0.31.

c) Because 75% lies within

\pm h

of the mean, the upper tail has probability

P(\bar X\ge168.6+h)=0.125.

d) Using a GDC,

168.6+h\approx171.867,

h\approx3.27\text{ cm}.

correctly explains the assumption based on the normality of the population distribution

[1 mark]

correctly determines the value of

\sigma_{\bar X}

[1 mark]

determines the probability

[1 mark]

correctly determines the probability in Question 14c)

[1 mark]

determines

h

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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