Test a revision-time claim and find a minimum sample size

A teacher claims that the average time a student spends on mathematics revision per day is 120 minutes. A random sample of 150 students is observed and the time, $x$ min, that each student spends on mathematics revision per day noted. It is found that

$\sum(x-120)=-1500,\qquad\sum(x-120)^2=462\,000.$

(i) Test, at the 1% level of significance, whether the teacher has overstated the average time spent on mathematics revision per day by a student.4 marks

(ii) Using the unbiased estimate of the population variance in (i), what is the least number of students that is required in a sample whose mean is 110 hours to arrive at a different conclusion from that in part (i)?2 marks