5 Amari is investigating how accurately people can estimate a short time period. He asks each of a random sample of 40 people to estimate a period of 20 seconds. For each person, he starts a stopwatch and then stops it when they tell him that they think that 20 s has elapsed. The times which he records are denoted by \(x \mathrm {~s}\). You are given that
\(\sum x = 765 , \quad \sum x ^ { 2 } = 15065\).

1. Determine a 95\% confidence interval for the mean estimated time.
2. Amari says that the confidence interval supports the suggestion that people can estimate 20 s accurately. Make two comments about Amari's statement.
3. Discuss whether you could have constructed the confidence interval if there had only been 10 people involved in the experiment. Amari thinks that people would be able to estimate more accurately if he gave them a second attempt. He repeats the experiment with each person and again records the times. Software is used to produce a \(95 \%\) confidence interval for the mean estimated time. The output from the software is shown below. Z Estimate of a Mean Confidence level 0.95 Sample

   Mean	19.68
   s	1.38
   N	40

   Result
   Z Estimate of a Mean

   Mean	19.68
   s	1.38
   SE	0.2182
   N	40
   Interval	\(19.68 \pm 0.4277\)

4. State the confidence interval in the form \(\mathrm { a } < \mu < \mathrm { b }\).
5. Make two comments based on this confidence interval about Amari's opinion that second attempts result in more accurate estimates.