2024 RVHS H1 Q11

An internet advertising company \(TicTakAim\) claims that viral video's duration has a mean duration of \(30\) seconds. An influencer wants to investigate the company's claim as she believes that the company is underestimating the mean duration. The influencer takes a random sample of \(90\) viral video's durations. The viral video's durations, \(x\) seconds, are summarised as follows.

\[\sum (x-30) = 90 \quad \quad \sum (x-30)^2 = 2037\]

1. Find unbiased estimates of the population mean and variance.[2]
2. Carry out an appropriate test, at the \(3\)% level of significance, whether the company's claim is justifiable. You should state your hypotheses and define any symbols you use.[4]
3. Explain, in the context of the question, the meaning of "at the 3% level of significance".[1]
4. State, giving a reason, whether it is necessary to assume that the durations of viral videos are normally distributed for the test to be valid.[1]

Assume now that the population standard deviation of viral videos' duration is 4.0 seconds. Another random sample of 50 durations of viral videos is taken. A test at the 3% level of significance supports TicTakAim's claim.

5. Find the range of values of the sample mean