2023 RI P2 Q8

From a school-wide survey done in 2022, it was found that the students in a certain school spend an average of \(45.5\) hours per week on social media platforms. In 2023, the school’s Student Well-being team decides to carry out a survey on a group of \(50\) randomly chosen students.

1. State what it means for a sample to be random in this context.[1]

The time, \(x\) hours, spent per week on social media platforms by the \(50\) students are summarised below.

\(\sum{\left( x-45.4 \right)}=55\)                             \({{\sum{\left( x-45.4 \right)}}^{2}}=\mathrm{1494}\mathrm{.74}\)

2. Calculate unbiased estimates of the population mean and variance of the time spent by students per week on social media platforms.[2]
3. State hypotheses that can be used to test if the mean time spent by students per week differs from \(45.5\) hours. Work out the critical region in this case, and use it to carry out the test at the \(5\%\) level of significance, giving your conclusion in the context of the question.[4]
4. Explain why it is not necessary to assume that the time spent by students per week on social media platforms follows a normal distribution when performing the test in part (c).[1]

It is now given that the time spent by students per week on social media platforms follows a normal distribution with a variance of \(25\) hours\(^{2}\). Based on observations, the Head of Student Well-being, Ms Tan, believes that the mean time spent by students per week on social media platforms has in fact increased.

5. By carrying out the test at \(\alpha %\) level of significance using data collected from \(50\) randomly chosen students with a mean of \(47\) hours, determine the set of values of \(\alpha \) for which Ms Tan’s belief should be accepted.[4]