• This assignment is important in providing feedback and helping to establish competency in essential skills.
• Answer all the questions. The questions are not of equal weight, and some questions are worth much more than others.
• The questions relate to material up to and including Module 10.
• Read the Introductory Material of the course before starting this Assignment.
• When you are asked to comment on a finding, usually a short paragraph is all that is required.
• For all graphs, label the axes correctly, include a contextual title and the units of measurement.
• In many cases, spss output contains much more information than is required for a correct and complete answer. In those cases just reproducing the output may not attract any marks. Make sure you report only the information from the spss output relevant to your answer.
• Unless instructed otherwise, show all working and formulae used in calculating confidence intervals and performing hypothesis tests. (Answers may of course be checked where possible using computer software.)
Question 1
Considering the two variables number of visits and number of adult materials in circula- tion in the data file Library2012New.sav, available on the course StudyDesk, assuming that the data represents an SRS, answer the following questions.
(a) Find estimates of the mean (μ) and standard deviation (σ) of the (i) number of visits and (ii) number of adult materials in circulation.
(b) Obtain a 90% confidence interval for the difference of means of the number of visits and number of adult materials in circulation in the libraries.
. (c) State the hypotheses to test, if the difference of means of the number of visits and number of adult materials in circulation is significant.
. (d) Find the value of the appropriate test statistic for the test in part (c).
. (e) Obtain the P-value, and make an appropriate conclusion on the outcome of the test.
. (f) What assumptions are necessary for the inference procedure in part (b) to be valid?
Note: In some parts of this question you may require to decide if a paired sample or two independent samples procedure is appropriate. You will not be penalised if you properly justify your choice and subsequent answers are correct.
Question 2
A new surgical procedure is successful with probability p = 0.8. Assume that the operation is performed five times and the results are independent of one another.
. (a) What is the probability that all five operations are successful?
. (b) What is the probability that less than two operations are successful?
. (c) Find the mean and standard deviation of number of successful operation.
. (d) If the procedure is performed 100 times, what is the probability that at least 95 operations are successful? [5 marks]
Question 3
A Psychology research team was interested to study the reaction time of university students. They took a random sample 100 students from across the university and administered a series of tests to determine the reaction time. The observed mean and standard deviation of the data are 27.35 seconds and 6.31 seconds respectively.
. (a) What is the sampling distribution of the sample mean? Justify your answer.
. (b) Find a 95% confidence interval for the mean reaction time of all university students.
(c) Give the correct interpretation of the above confidence interval.
. (d) Calculate the margin of error for a 99% confidence interval for the mean reaction time. What is the width of the 99% confidence interval?
. (e) If the population standard deviation is 8 seconds, what sample size would be required to produce a 95% confidence interval for the population mean reaction time with a margin of error of 1.50 seconds?
Question 4
From long term experience it is known that the time required to answer a set of 10 computer managed questions in the Data Analysis course follows a normal distribution with mean μ = 15 minutes and standard deviation σ = 2 minutes. If a randomly chosen off-campus student answers a test of 10 computer managed questions, answer the following questions.
. (a) What is probability that she would complete the test in less than 14 minutes?
. (b) What is the probability that she would complete the test between 15 and 19 minutes?
. (c) Determine her completion time so that only 10% of the students doing the test will take longer than her.
. (d) For a set of 5 randomly selected tests (each with 10 questions), what is the probability that her mean completion time will be 14 minutes or more?
Question 5
In January this year, 200 randomly selected voters in Australia were asked whether they believed that the Government is doing a good job to protect the environment.
. (a) If 156 of these 200 voters believe the Government is doing a good job, deter- mine a 90% confidence interval for the true proportion of voters who believe the Government is doing a good job to protect the environment.
. (b) In previous years, approximately 70% of the the voters believed the Government was doing a good job to protect the environment. Has the proportion of voters who believe the Government is doing a good job to protect the environment changed? Test this hypothesis at the 1% level. Show all your working.
Question 6
Answer the following questions:
. (a) In no more than 100 words, identify the problems associated with non-random sampling.
. (b) Based on a random sample of size n = 144 from a population with proportion p = 0.52, explain the sampling distribution of the sample proportion. State the name of the distribution, underlying parameter(s), and any assumptions required.
. (c) Based on a random sample of size n = 144 from a population with mean μ = 20 and standard deviation σ = 6, explain the sampling distribution of the sample mean. In your answer you may state the name of the distribution, underlying parameter(s), and any assumptions required.
. (d) State how you would describe any association between (i) two categorical variables, (ii) one categorical and one quantitative variable, and (iii) two quantitative variables.
. (e) State and explain the Central Limit Theorem (CLT) for the sample mean when the population distribution is (i) symmetric and (ii) not symmetric.
. (f) With appropriate examples, distinguish between paired samples and two independent samples.