| Date | May 2017 | Marks available | 4 | Reference code | 17M.1.hl.TZ0.13 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Show that and Hence | Question number | 13 | Adapted from | N/A |
Question
A random sample \({X_1},{\text{ }}{X_2},{\text{ }} \ldots ,{\text{ }}{X_n}\) is taken from the normal distribution \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\), where the value of \(\mu \) is unknown but the value of \({\sigma ^2}\) is known. The mean of the sample is denoted by \(\bar X\).
A mathematics teacher, wishing to apply the above result, generates some artificial data, assumes a value for the variance and calculates the following 95% confidence interval for \(\mu \),
\[[3.12,{\text{ }}3.25].\]
The teacher asks Alun to interpret this result. Alun makes the following statement. “The value of \(\mu \) lies in the interval \([3.12,{\text{ }}3.25]\) with probability 0.95.”
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.
Markscheme
\(\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\) 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]
