7 An ambulance control centre responds to emergency calls in a rural area. The response time, \(T\) minutes, is defined as the time between the answering of an emergency call at the centre and the arrival of an ambulance at the given location of the emergency.
Response times have an unknown mean \(\mu _ { T }\) and an unknown variance.
Anita, the centre's manager, asked Peng, a student on supervised work experience, to record and summarise the values of \(T\) obtained from a random sample of 80 emergency calls.
Peng's summarised results were
$$\text { Mean, } \bar { t } = 6.31 \quad \text { Variance (unbiased estimate), } s ^ { 2 } = 19.3$$
Only 1 of the 80 values of \(T\) exceeded 20
- Anita then asked Peng to determine a confidence interval for \(\mu _ { T }\). Peng replied that, from his summarised results, \(T\) was not normally distributed and so a valid confidence interval for \(\mu _ { T }\) could not be constructed.
- Explain, using the value of \(\bar { t } - 2 s\), why Peng's conclusion that \(T\) was not normally distributed was likely to be correct.
- Explain why Peng's conclusion that a valid confidence interval for \(\mu _ { T }\) could not be constructed was incorrect.
- Construct a \(98 \%\) confidence interval for \(\mu _ { T }\).
- Anita had two targets for \(T\). These were that \(\mu _ { T } < 8\) and that \(\mathrm { P } ( T \leqslant 20 ) > 95 \%\).
Indicate, with justification, whether each of these two targets was likely to have been met.
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