Tung Wah College GED1008 Introduction to Statistics Assignment 2 [Deadline: 19 March 2021 Friday, 11:59pm] Instructions 1. Answer all questions. Write your answers on standard A4 papers. Do not type. 2. Submit the softcopy as one PDF file in the blackboard. You can scan your assignment using any Mobile Phone Apps, such as Adobe Scan, FastScanner, which can save your file as a single PDF. 3. Write your steps clearly. Answers without any steps will have no marks. 4. Make reasonable rounding unless the rounding rule is explicitly stated in the question. 5. Make sure that all your answers are written in proper English. Marks could be deducted for any English or grammatical mistakes. 6. No plagiarism is tolerated. All plagiarized work will score 0. Question 1 (17 marks) A research is performed to study the time (in hours) teenagers spent on study and physical exercises in a week. A literature reported the population mean (𝜇), the population standard deviation (𝜎) and the distribution shape of these two variables. The table below gives a summary. Variable 𝜇 𝜎 Distribution shape Time spent on study 23.8 3.1 Normal distribution Time spent on physical exercises 18.5 2.3 Left-skewed distribution Based on the data above, answer the following questions. a) b) c) d) e) Find the percentage of teenagers who spend more than 28 hours on study in a week. [2] Suppose the researcher selects a teenager from the population in random. Find the probability that the teenager spends less than 21 hours on study in a week. [2] The researcher wants to identify teenagers whose time spent on study are at the bottom 7.5%. What is the time spent below which teenagers will be identified? [3] Find the minimum proportion of teenagers who spend between 15 and 22 hours on physical exercises in a week. Can we conclude that more than 50% of teenagers will spend more than 15 hours on physical exercises in a week? (Hints: note that the time spent on physical exercises does not follow the normal distribution and the standard normal distribution table cannot be used). [3] A sample of 50 teenagers is selected from the population. Find the probability that the sample mean time spent on physical exercises is higher than 19.5 hours in a week. Do you think it is 1 likely to obtain such a sample? [3] f) g) What can you say about the population mean time spent on physical exercises by teenagers reported by the literature if the sample in (e) is really obtained? [2] Do you think your answers in (e) will be the same if the time spent on physical exercises is “rightskewed”? Explain your answer briefly. [2] Question 2 (13 marks) A research is performed to investigate the mental health of Hong Kong residents during the COVID19 pandemic and their willingness to get COVID-19 vaccination. The mental health is measured using “Patient Health Questionnaire 9” (PHQ-9), with a scale from 0 to 27, with a higher score meaning more severe depression. From a sample of 61 Hong Kong residents, the PHQ-9 scores are measured. They are also asked about their willingness to get COVID-19 vaccination. Data collected are summarized in the tables below. Table 1: The PHQ-9 scores of a sample of 61 Hong Kong residents Sample Mean Sample Standard Deviation 6.8 4.1 Table 2: Are you willing to get COVID-19 vaccination when it is available? Yes No 19 42 a) Find the 99% confidence interval for the population mean for data in Table 1. Write the conclusion in the context of the subject matter. [5] b) According to the PHQ-9 instruction, when the PHQ-9 score reaches 5, the participant has mild depression severity. Is it reasonable to conclude that, on average, Hong Kong residents have mild depression severity? Explain your answer. [2] c) Find the 95% confidence interval for the population proportion of Hong Kong residents who are willing to get COVID-19 vaccination. Write the conclusion in the context of the subject matter. [4] d) Can it be concluded that more than 35% of all Hong Kong residents are willing to get COVID-19 vaccination? Explain your answer. [2] 2 Instructions 1. Answer all questions. Write your answers on standard A4 papers. Do not type. 2. Submit the softcopy as one PDF file in the blackboard. You can scan your assignment using any Mobile Phone Apps, such as Adobe Scan, FastScanner, which can save your file as a single PDF. 3. Write your steps clearly. Answers without any steps will have no marks. 4. Make reasonable rounding unless the rounding rule is explicitly stated in the question. 5. Make sure that all your answers are written in proper English. Marks could be deducted for any English or grammatical mistakes. 6. No plagiarism is tolerated. All plagiarized work will score 0. Question 1 (17 marks) A research is performed to study the time (in hours) teenagers spent on study and physical exercises in a week. A literature reported the population mean (𝜇), the population standard deviation (𝜎) and the distribution shape of these two variables. The table below gives a summary. 𝜇 Variable 𝜎 Distribution shape Time spent on study 23.8 3.1 Normal distribution Time spent on physical exercises 18.5 2.3 Left-skewed distribution Based on the data above, answer the following questions. a) b) Find the percentage of teenagers who spend more than 28 hours on study in a week. [2] Suppose the researcher selects a teenager from the population in random. Find the probability that the teenager spends less than 21 hours on study in a week. [2] c) The researcher wants to identify teenagers whose time spent on study are at the bottom 7.5%. What is the time spent below which teenagers will be identified? [3] d) Find the minimum proportion of teenagers who spend between 15 and 22 hours on physical exercises in a week. Can we conclude that more than 50% of teenagers will spend more than 15 hours on physical exercises in a week? (Hints: note that the time spent on physical exercises does not follow the normal distribution and the standard normal distribution table cannot be used). [3] e) A sample of 50 teenagers is selected from the population. Find the probability that the sample mean time spent on physical exercises is higher than 19.5 hours in a week. Do you think it is 1 likely to obtain such a sample? [3] f) g) What can you say about the population mean time spent on physical exercises by teenagers reported by the literature if the sample in (e) is really obtained? [2] Do you think your answers in (e) will be the same if the time spent on physical exercises is “rightskewed”? Explain your answer briefly. [2] Question 2 (13 marks) A research is performed to investigate the mental health of Hong Kong residents during the COVID19 pandemic and their willingness to get COVID-19 vaccination. The mental health is measured using “Patient Health Questionnaire 9” (PHQ-9), with a scale from 0 to 27, with a higher score meaning more severe depression. From a sample of 61 Hong Kong residents, the PHQ-9 scores are measured. They are also asked about their willingness to get COVID-19 vaccination. Data collected are summarized in the tables below. Table 1: The PHQ-9 scores of a sample of 61 Hong Kong residents Sample Mean Sample Standard Deviation 6.8 4.1 Table 2: Are you willing to get COVID-19 vaccination when it is available? Yes No 19 42 a) Find the 99% confidence interval for the population mean for data in Table 1. Write the conclusion in the context of the subject matter. [5] b) According to the PHQ-9 instruction, when the PHQ-9 score reaches 5, the participant has mild depression severity. Is it reasonable to conclude that, on average, Hong Kong residents have mild depression severity? Explain your answer. [2] c) Find the 95% confidence interval for the population proportion of Hong Kong residents who are willing to get COVID-19 vaccination. Write the conclusion in the context of the subject matter. [4] d) Can it be concluded that more than 35% of all Hong Kong residents are willing to get COVID-19 vaccination? Explain your answer. [2] 2
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