Assignment 2 Semester 1, 2013 This assignment has four questions, and is due by 5. 00pm on Thursday 2 May. It is to be submitted electronically as a . pdf ? le using the assignment tool on the subject’s LMS page. Marks depend on your tutor being able to understand your statements and arguments, so marks may be deducted for poor presentation or unclear language. Use nothing smaller than 12 point font. If you wish to write your assignment by hand and scan the ? le into a . pdf format, you may, though any illegible content will not be marked.
You may work in groups of up to 4 students from the same allocated tutorial. Groups should nominate one student to submit the assignment for the whole group, with each student’s name and student number included in the document. This assignment has a total of 40 marks available and may contribute up to 10% of your ? nal mark in this subject. Question 1 (5 marks) Recall the concept of the sampling error from the ? rst and second weeks of semester. In lectures in Week 6, we considered an example in which Optus was estimating the mean telephone expenditure of a household. In the context of that example, answer each of the following questions.
Where calculations are required, show all your workings. (a) Suppose we know from experience that the population standard deviation is 1200, and we want a 90% con? dence interval for the population value of annual telephone expenditure with an error of estimation (ie the sampling error) no greater than $50. What sample size should we use? (3 marks) (b) Suppose we want a 90% con? dence interval for the population value of annual telephone expenditure that is no wider than $100. What sample size should we use? (1 mark) (c) Explain in words what parts (a) and (b) tell us about the relationship between sampling error and the width of a con? ence interval estimate. (1 mark) 1 Question 2 (21 marks) You are consulting for a farmer who grows and supplies millions of tomatoes to a supermarket chain. The farmer does not weigh every single tomato produced. In 2012, the farmer was told by the supermarket that the tomatoes she supplies to them have an average weight of 74 grams. It is well known that the weight of tomatoes is normally distributed about their mean. The farmer also knows that the supermarket chain has a policy of rejecting tomatoes that are too small. Speci? cally, they reject (and will not pay for) tomatoes that weigh less than 70 grams.
Given this information, answer the following questions. Where calculations are required, show all your workings. (a) The supermarket manager has told the farmer that 90. 99% of all her tomatoes produced in 2013 have been accepted. What is the standard deviation of tomatoes grown by the farmer in 2013, in grams? (3 marks) The farmer is worried that the fact that only around 9 out of 10 of her tomatoes are accepted might suggest that the average weight of her tomatoes has fallen below 74 grams. You take a random sample of 70 tomatoes, with a mean of 72 grams and a standard deviation of 9 grams. (b) Test, at the 5% level of signi? ance, the hypothesis that the mean weight of the farmer’s tomatoes has fallen below it’s 2012 value. (5 marks) (c) Do the results of the test in part (b) prove with certainty whether the null hypothesis is true or false? Explain why or why not. (2 marks) (d) Test, at the 1% level of signi? cance, the hypothesis that there is the mean weight of the farmer’s tomatoes has fallen below it’s 2012 value. (5 marks) (e) Note that the results of your tests in parts (b) and (d) are contradictory. The farmer is unimpressed by this, and claims that if you were doing your job properly, you wouldn’t be able to draw two di? rent conclusions from the same set of data. How can you explain to the farmer that it is possible to draw these two di? erent conclusions from the same data? (2 marks) (f) Calculate the p-value for the hypothesis tests in parts (b) and (d). How might knowing this value improve the explanation given to the farmer in part (e)? (4 marks) Hint: Think about the de? nition of the p-value and how this might let you provide a more concise and speci? c response to the farmer’s concerns Question 3 (6 marks) You are considering entering the market for bikes as a seller.
If your bikes are of higher quality than the average bike already in the market for bikes, you will make a 2 pro? t by entering that market. If your bikes are not of higher quality than the average bike already in the market, you will make a loss by entering the market for bikes. You test, at the 5% level of signi? cance, the hypothesis that the quality of your bikes exceeds the average quality of existing bikes in the market. Given this information, answer the following questions. (a) What would a Type I Error be in the context of this case? Explain what the consequences of a Type I Error would be. 2 marks) (b) What would a Type II Error be in the context of this case? Explain what the consequences of a Type II Error would be. (2 marks) (c) After conducting the test, you conclude that there is su? cient evidence in the data you observe to support the alternative hypothesis that your bikes are of higher quality, and so decide to enter the market for bikes as a seller. Suppose that you are particularly concerned about the risk of making a Type I Error. Which of the following methods would reduce the chance of doing so, and why? (2 marks) (i) Repeating the test once with a smaller level of signi? ance. (ii) Repeating the test once with a larger sample size. (iii) Repeating the same test many times to get more information. (iv) Both (i) and (ii), but not (iii). (v) Both (ii) and (iii), but not (i). Question 4 (8 marks) An important part of university study involves extending the application of ideas you have learned to new contexts. We know that the sample mean has a sampling distribution which we can exploit to construct con? dence interval estimates of that population parameter. Chapter 15 of the textbook discusses inference about population variances, rather than about the population mean.
After reading this chapter answer the following question. Suppose that you are the quality control manager for a company that manufactures fetilizer. Part of your job involves keeping track of the level of impurities in the fertilizer. The standard required is that in 40kg bags of fertilizer the variance in the kilograms of impurities should not exceed 1kg2 . (a) Suppose that a random sample of 20 bags of fertilizer is obtained and the quantity of impurities in each bag is measured. The sample variance is computed to be s2 = 3kg2 . Construct a 95% con? dence interval for the population variance of fertilizer impurities. 2 marks) (b) Using the information in part (a), test the hypothesis that the population variance does not exceed 1kg2 against the alternative that it does. (3 marks) 3 (c) A market watchdog decides to compare the quality of your fertilizer with that of another manufacturer. It does this by taking a further sample of 20 40kg bags of fertilizer from the other maufacturer and ? nds that the variance of impurities in their product is s2 = 2. 85kg2 . Test the proposition that your product is not significantly di? erent from that of the other manufacturer, in terms of quality control, against the alternative that they are. (3 marks) 4