STAM4000 QUANTITATIVE METHODS Assignment 2 – 2013 Trimester 1 General guidelines for submission Answer all questions, clearly show your workings and include relevant graphs or statistical output to...

STAM4000 QUANTITATIVE METHODS


Assignment 2 – 2013 Trimester 1


General guidelines for submission

• Answer all questions, clearly show your workings and include relevant graphs or statistical output to support your answers.


• Clearly show all formula to support your answers.


• You may use Microsoft Excel or any other statistical software. Please copy and paste any relevant output into the Microsoft word document.

Question 1 (18 marks)

An entire population of English speaking adults living on a remote island named Statistica underwent IQ testing and the recorded IQ scores are presented in the file QMAssignment 2.xls.

1. Examine the data and determine if the distribution of IQ scores could be normal. You should confirm or reject normality by means of a visual inspection of the shape of the distribution.

Assume now that the distribution of IQ scores in Statistica is indeed normal with parameters equal to the mean and standard deviation of the given population data.

1. What proportion of the population falls into the “gifted” category where gifted is defined as an IQ score in the range 130 to 144?

1. What proportion of the population falls into the “genius” category where genius is defined as an IQ score above 144?

1. An alternative definition of “genius” is someone with an IQ score above the 99.9^th
   percentile score. Under this definition, what is the minimum IQ score to be classified as a genius?

Suppose that a random sample of 150 Statistica residents is selected.

1. What is the probability that the average IQ of this sample is below 95?

Use Excel or other software to obtain an exact probability value.

1. What is the probability that the average IQ of this sample is above 144?

Use Excel or other software to obtain an exact probability value.

1. Explain briefly why the probabilities calculated in parts (c) and (f) differ?

Question 2 (20 marks)

Part A
A manufacturer of LCD panel televisions claims the mean lifetime of its screens is 58,000 hours. Assume from historical experience that the lifetime of the screens is normally distributed with a known variance of 25 million (hours)².
In order to test the manufacturer’s claim, an independent consumer protection organisation randomly acquired 30 of these televisions and operated them until the screens failed. The mean lifetime was measured to be 56,475 hours.

1. Construct a 95% confidence interval for the true lifetime of the screens.

1. Is there any reason to challenge the claim of the manufacturer?

1. An individual customer’s television screen failed after 52,500 hours. Assuming that the true mean lifetime is 58,000 hours, what is the probability of this occurring?

Part B
The University of Brisbane asked its first year students how long they had lived at their current address. The results of 797 students had a median of 6 years, a mean of 8.8 years and a standard deviation of 6.4 years.

1. Is it likely that the population distribution for the length of time at current address is normal? Explain your answer.

1. Given the answer above, can a 97% confidence interval be determined for the population mean? Explain why it can or cannot be determined.

1. Assume now that the confidence interval can validly be determined. Construct the 97% confidence interval for the population mean.

1. What would happen to the width of the confidence interval if there were greater variability in the length of time at current address?

Part C
A recent poll asked 620 randomly selected Australians if they preferred Julia Gillard or Tony Abbott as prime minister. The results showed that 217 of those surveyed nominated Julia Gillard as the preferred option.

1. Calculate a 95% confidence interval for the proportion of the population that prefers Julia Gillard.

1. If the pollster wanted to reduce the margin of error to 2%, what would be the required sample size to achieve this increased level of accuracy? Use the most conservative assumption about the sample proportion.

Question 3 (12 marks)

Your employer is a medium size IT company that produces smartphones and tablets. Your sales director approaches you with concerns about possible declining sales in the face of stiff competition from other larger producers such as Apple and Samsung who launched new products at the beginning of the year. Recent sales figures for the last 26 weeks are shown in the file QMAssignment 2.xls.
You have been asked to provide statistical evidence concerning the claim that recent weekly sales have declined following the launch of your competitor’s new devices. Prior to the beginning of the year, it was accepted that your company had mean weekly sales of $135,000.

1. Specify the most appropriate choice of null and alternative hypotheses to test whether there has been a decline in mean weekly sales.

1. Provide the formula for the test statistic and compute its value.

1. What is the critical value of the test using a significance level of 2.5 per cent?

1. What is the p-value of the test? Use Excel to obtain an exact value.

1. What is your statistical decision regarding the pair of hypotheses?

1. What is your conclusion (in non-statistical language) about mean weekly sales?

1. Repeat parts (c), (e) and (f) using a significance level of 0.5 per cent.

Question 4 (24 marks)

In this question, we will be using information about attendances at Australian Rules Football (AFL) matches held at the Melbourne Cricket Ground (MCG). The data is in the Excel file: QM Assignment 2.xls. A description of the variables is in the Excel file.
[Data obtained from: Smyth, GK (2011). Australasian Data and Story Library (OzDASL). http://www.statsci.org/data.]
Use Excel (or other software) to estimate the following linear regression model:

1. Write out the estimated regression equation.

1. Interpret the estimated intercept.

1. Which variables are statistically significant predictors of match attendance at the MCG? Use a significance level of 5%.

1. Interpret the estimated coefficient for the temperature variable.

1. Interpret the estimated coefficient for the non-Melbourne team variable.

1. Interpret the F-statistic.

1. Comment on the goodness of fit of the estimated model.

1. Predict the attendance at a match between Adelaide (25,000 members) and the Western Bulldogs (16,500 members) who have a total of 4 Top 50 players between them, on a day with a forecast temperature of 14 degrees Celsius and a forecast attendance of 65,000 at the other matches in Melbourne.

1. Carry out an analysis of the model’s residuals to investigate whether the model may have a heteroskedasticity problem. What would be the implication if there were a heteroskedasticity problem?

1. How do you think the above model could be improved?