2013 AJC H1 Q9

A teacher of Peaceful Childcare Centre claims that a child at the centre spends an average of 4 hours on afternoon nap. A survey was conducted and the time, \(x\) hours, spent by \(50\) randomly chosen children on afternoon naps is as follows:

\(\sum{x}=185\), \(\sum{{{\left( x-\bar{x} \right)}^{2}}}=39.2\)

1. Find an unbiased estimate of the population mean and variance.[2]
2. Carry out a test at \(2\%\) level of significance to determine if the teacher’s claim is correct.[4]
3. Suppose the teacher now claims that a randomly chosen child spends less than \(4\) hours on afternoon nap. Without carrying out another test, state, with a reason, whether the conclusion in part (ii) would remain the same.[2]

The principal of Peaceful Childcare Centre now claims that a randomly chosen child spends \({{\mu }_{0}}\) hours on afternoon nap. Find the set of values of \({{\mu }_{0}}\) such that the principal is confident, at \(5\%\) level of significance, that he did not understate the mean time that the children took for afternoon naps.[4]