2024 HCI P2 Q10

2024 HCI P2 Q10

12 marks

An online website, Star-Salary, which shares information on the salaries for fresh graduates in Singapore, claimed that the mean monthly salary of a fresh graduate with a Bachelor of Science (B.Sc) degree was \(\$3600\).

However, another website, First-Pay, stated a higher mean monthly salary for a fresh graduate with the same degree. A random sample of \(80\) fresh graduates with a B.Sc degree is surveyed and their monthly salaries, \(\$x\), are summarized by

\[\sum (x - 3600) = 1000, \qquad \sum (x - 3600)^2 = 205000\]

  1. Give a reason why it is challenging to obtain a random sample in this context.[1]
  2. Calculate exact unbiased estimates of the population mean and variance for the monthly salaries of fresh graduates with a B.Sc degree.[2]
  3. Test, at the \(5\%\) level of significance, whether First-Pay's claim is justified. You should state your hypotheses and define any parameters that you use.[4]

  4. Explain, with justification, whether any assumption about the population is needed for the test in part (c) to be valid.

    [1]
  5. State, in the context of the question, the meaning of "\(5\%\) level of significance".[1]

A second sample of \(60\) randomly chosen fresh graduates with B.Sc degree is surveyed and the sample mean and standard deviation of their monthly salaries are found to be \(\$\bar{y}\) and \(\$355\) respectively.

  1. Find the largest value of \(\bar{y}\) such that this second sample would conclude the test in favour of Star-Salary's claim at \(5\%\) level of significance, giving your answer correct to the nearest dollar.[3]
Similar questions are unavailable for this question.
Answer:(a) Not all graduates will be employed immediately after graduation, making it harder to gather salary data and select a truly random sample. It may be difficult to track where fresh graduates are employed, as their contact details may have changed since leaving university. (b) \({\bar{x} = \frac{7225}{2}}\), \({s^2 = 2436.708861}\) (c) At \({5\%}\) level of significance, \({p\text{-value} \approx 0.0118}\) (3 s.f.) Since \({p\text{-value} = 0.0118 < 0.05}\), we reject \({H_0}\) and conclude that there is sufficient evidence at \({5\%}\) level of significance that the population mean monthly salary is higher than \({\$3600}\). Therefore First-Pav's claim is justifiable. (d) Since the sample size of \({80}\) is large, by Central Limit Theorem, sample mean monthly salary of fresh graduates with B.Sc, \({\overline{X}}\) follows a normal distribution approximately. Thus, no assumption on the population, \({X}\) is needed. (e) "\({5\%}\) level of significance" means that there is a \({5\%}\) probability that we wrongly conclude that population mean monthly salaries of fresh graduates with B.Sc. is higher than \({\$3600}\) when it is in fact \({\$3600}\). (f) Largest value of \({\bar{y} = 3676}\) (nearest dollar)

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