State the distribution of $\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}$.

Hence show that, with probability 0.95,

$$\bar X - 1.96\frac{\sigma }{{\sqrt n }} \leqslant \mu  \leqslant \bar X + 1.96\frac{\sigma }{{\sqrt n }}.$$

Explain briefly why this is an incorrect statement.

Give a correct interpretation.

$\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}$ is ${\text{N}}(0,{\text{ }}1)$ or it has the Z-distribution A1

[??? marks]

attempt to make a probability statement     R1

therefore with probability 0.95,

$- 1.96 \leqslant \frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}} \leqslant 1.96$     A1

$- 1.96\frac{\sigma }{{\sqrt n }} \leqslant \bar X - \mu  \leqslant 1.96\frac{\sigma }{{\sqrt n }}$     A1

$1.96\frac{\sigma }{{\sqrt n }} \geqslant \mu  - \bar X \geqslant  - 1.96\frac{\sigma }{{\sqrt n }}$     A1

$\bar X + 1.96\frac{\sigma }{{\sqrt n }} \geqslant \mu  \geqslant \bar X - 1.96\frac{\sigma }{{\sqrt n }}$

Note:     Award the final A1 for either of the above two lines.

$\bar X - 1.96\frac{\sigma }{{\sqrt n }} \leqslant \mu  \leqslant \bar X + 1.96\frac{\sigma }{{\sqrt n }}$     AG

[??? marks]

you cannot make a probability statement about a constant lying in a constant interval OR the mean either lies in the interval or it does not     A1

[1 mark]

the confidence interval is the observed value of a random interval

OR if the process is carried out a large number of times, $\mu$ will lie in the interval 95% of the times     A1

[1 mark]