a) Suppose that Y₁, Y₂, ..., Y₁ is a random sample from a normal distribution with parameter mean 0 and variance 8, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects Ho if y ≥c, where c is a constant that solves the probability equation 0.05 = P(Y≥c|0 = 1). ii) What will be the value of c in part a) if n = 15. b) An academic Professor conducted a research aimed at exploring the experience and challenges faced by people who are diagnosed with dementia and their families. If the professor perform a hypothesis test to decide whether the mean age of all people diagnosis with initial occurrence of dementia is less than 55 years old. Based on the statement, explain, what each of the following would mean. i) Type I error. ii) Correct decision. iii) Type II error. c) Suppose that the results of carrying out the hypothesis test lead to non-rejection of the null hypothesis. Classify that conclusion by error type or as a correct decision if in fact the mean age at diagnosis of all people with early-onset dementia is less than 55 years old. Suppose that the manufacturer of a new COVID-19 vaccine in South Africa wants to test the null hypothesis Ho: 0= 0.90 against the alternative hypothesis H₁:0=0.60. She defined her test statistic Y as a number of patients recovered from COVID-19 after an observed twenty trials. If she decided to accept H, and assume y > 14, i) What will be her Type I error and the power of the test. ii) Explain how the probability of Type II error can be reduced in this case and how it will affect Type I error. Give a practical example. d) A South African manufacturer of BMW cars offers a model in one of three colors: Blue, Red, or Black. In his report, the manufacturer indicated that of the first 1000 cars sold, 400 were of color Blue. With this information from the manufacturer, perform a test at a = 1% and conclude whether or not the customers do have more preference for BMW cars which have Blue color? Show all necessary steps and justify your answer.

a) Suppose that Y₁, Y₂, ..., Y₁ is a random sample from a normal distribution with parameter mean 0 and variance 8, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects Ho if y ≥c, where c is a constant that solves the probability equation 0.05 = P(Y≥c|0 = 1). ii) What will be the value of c in part a) if n = 15. b) An academic Professor conducted a research aimed at exploring the experience and challenges faced by people who are diagnosed with dementia and their families. If the professor perform a hypothesis test to decide whether the mean age of all people diagnosis with initial occurrence of dementia is less than 55 years old. Based on the statement, explain, what each of the following would mean. i) Type I error. ii) Correct decision. iii) Type II error. c) Suppose that the results of carrying out the hypothesis test lead to non-rejection of the null hypothesis. Classify that conclusion by error type or as a correct decision if in fact the mean age at diagnosis of all people with early-onset dementia is less than 55 years old. Suppose that the manufacturer of a new COVID-19 vaccine in South Africa wants to test the null hypothesis Ho: 0= 0.90 against the alternative hypothesis H₁:0=0.60. She defined her test statistic Y as a number of patients recovered from COVID-19 after an observed twenty trials. If she decided to accept H, and assume y > 14, i) What will be her Type I error and the power of the test. ii) Explain how the probability of Type II error can be reduced in this case and how it will affect Type I error. Give a practical example. d) A South African manufacturer of BMW cars offers a model in one of three colors: Blue, Red, or Black. In his report, the manufacturer indicated that of the first 1000 cars sold, 400 were of color Blue. With this information from the manufacturer, perform a test at a = 1% and conclude whether or not the customers do have more preference for BMW cars which have Blue color? Show all necessary steps and justify your answer.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 6CR

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