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**Question 1**

An education researcher wanted to examine whether there was a difference in academic performance between students who did face-to-face lessons and those who did online lessons.

To achieve this, he recorded the overall scores (upon 100%) of a class of 45 SUSS students who studied Statistics through face-to-face mode during the 2019 July semester. Similarly, he took the overall scores of another class of 50 students who studied Statistics through the online mode during the 2020 July semester. The dataset can be found in the TOA Data File ‘Question_1’ worksheet.

(a) Describe three (3) measures of location and three (3) measures of dispersion. State the values of these measures for both classes. You are required to present the relevant Excel reports.

(b) Show three (3) statistical concerns on the approach used in the study. If you have a free hand to study this research question starting from data collection to analysis, illustrate how you would conduct the project.

**Question 2**

(a) Mrs. Reena has regularly contacted three companies (A, B, and C) for her quarterly servicing of aircon. If A services her aircons, the probability that there will be no problem with the aircon in the next quarter is 0.8. The same probability for B is 0.7 and for C is 0.6. If all her aircon does not show any problems, she will engage the same company in the next quarter. If any of the aircons shows a problem, then she will engage A with a probability of 0.5, B with a probability of 0.3 or C with a probability of 0.2 in the next quarter. One of her aircons had a problem last quarter.

(i) Identify the probability that her aircon will have no problem in the next quarter.

(ii) Given that her aircon had no problem in the next quarter, what is the the probability that she engaged C?

(iii) What is the probability that she will engage A for the next two quarters?

(b) A study showed that the length of human pregnancies is normally distributed with a mean of 266 days and a standard deviation of 16 days.

(i) If a pregnant woman is randomly selected, identify the probability that her length of pregnancy is between 260 and 270 days.

(ii) If twenty (20) women are randomly selected, report the probability that the mean length of pregnancies is greater than 265 days.

**Question 3**

(a) A Psychology student wanted to study whether there is a difference in stress levels between gender. She devised a survey to measure stress. There were 20 questions with a total score of 100. She randomly selected 30 male and 30 female students from SUSS to complete the survey. The stress scores of these 60 students can be found in the TOA Data File ‘Question_3a’ worksheet.

(i) Describe the 90% confidence intervals for the mean stress score of each gender. Based on the confidence intervals, discuss whether we can conclude that the stress level of both genders is similar. You are required to present the relevant Excel reports.

(ii) Apply an appropriate hypothesis test at the 10% significant level to determine whether the stress levels of both genders are similar. You are required to present the relevant Excel report.

(b) The Psychology student extended her study to determine whether there is a difference in stress levels among SUSS students, SIM students, NUS students, and NTU students. She utilized the same instrument to measure stress. The stress scores of samples of students from the four universities were in TOA Data File ‘Question_3b’ worksheet. Conduct an appropriate hypothesis test, at the 5% significant level. and interpret whether there is a difference in the mean stress scores among the four universities.

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