5 Students at two colleges, \(A\) and \(B\), are competing in a computer games challenge.
- The time taken for a randomly chosen student from college \(A\) to complete the challenge has a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 10 students chosen from college \(A\). The results are summarised as follows.
$$\sum x = 828 \quad \sum x ^ { 2 } = 68622$$
A test is carried out on the data at the \(5 \%\) significance level and the result supports the claim that \(\mu > k\).
Find the greatest possible value of \(k\).
- A random sample of 8 students is chosen from college \(B\). Their times to complete the same challenge give a sample mean of 79.8 minutes and an unbiased variance estimate of 9.966 minutes \({ } ^ { 2 }\).
Use a 2 -sample test at the \(5 \%\) significance level to test whether the mean time for students at college \(B\) to complete the challenge is the same as the mean time for students at college \(A\) to complete the challenge. You should assume that the two distributions are normal and have the same population variance.