Snakes Are Commonly Encountered In The Australian Bush, Snakes Are Venomous, Alternative Hypothesis - Engineering Assignment Help

Assignment Task -                 
 

Problem 1 
(a.) Snakes are commonly encountered in the Australian bush, with about 15% of the snakes encountered being venomous. 

(b.) Suppose that during a bush walk you encounter 12 snakes. What is the probability that at least one of them is venomous? Carefully state your assumptions. 

(c.) What is the probability that between 2 to 4 (out of 12) snakes are venomous? 

What is the probability that the 10th snake you encounter during your walk is the 3rd one that is venomous? 
Problem 2 
(a.) A light bulb manufacturing company has lately received complaints about the lifetime of their product being too short. They have hired you to investigate if this is the case and, if so, they will invest a significant amount of money in overseeing the production. You may assume that the life-time of a light bulb follows a normal distribution with a standard deviation of 30. 

(b.) The reason the customers complained is that they expect that the mean lifetime of a light bulb is at least 300 hours. Write down the null and alternative hypothesis that you would test to carry out the task the company has assigned you. 

(c.)Given your hypothesis in (a.), can you explain in words the two different errors that may result from your investigation? Which of these errors do you think is worse to commit in this case? Explain. You want to ensure that you will take a sample size large enough to conduct a reliable hypothesis test. Given that the company will invest a significant amount of money if you conclude that the life-time is too short, you want a probability of 0.01 to falsely conclude that the life-time is too short when in fact it is not. If the lifetime is as short as 270 hours, you want your test to have a probability of 0.999 to detect this, because the company may lose many customers if they sell a defective product. How many observations should be included in your sample to meet these requirements? Carefully state the Type I and II error probabilities implied from the question. 

(d.) You obtain n = 30 measurements of lifetimes, with x¯ = 293.897 and s = 29.85687. Test your hypothesis in (a.) using the significance level implied from your answer in (c.). What is your advice to the manufacturing company? Carefully state your assumptions. 

(e.) During a meeting to present your results to the board of the manufacturing company, you are asked to provide an interval for the mean life-time of their light bulb product. Provide such an interval and interpret it in words for the board. Express yourself using non-technical terms. Can you use this interval to test the hypothesis in (a.)? Explain why/why not. 

(f.) Suppose that a future customer buys a light bulb. Give an interval for the light bulb lifetime that this customer can expect. 

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