IQ test

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Question 1 (18 marks)

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

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a) 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. [2 marks]

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

b) What proportion of the population falls into the “gifted” category where gifted is defined as an IQ score in the range 130 to 144? [3 marks]

c) What proportion of the population falls into the “genius” category where genius is defined as an IQ score above 144? [2 marks]

d) An alternative definition of “genius” is someone with an IQ score above the 99.9th percentile score. Under this definition, what is the minimum IQ score to be classified as a genius? [3 marks]

Suppose that a random sample of 150 Statistica residents is selected.
e) 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.[3 marks]

f) 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.[3 marks]

g) Explain briefly why the probabilities calculated in parts (c) and (f) differ? [2 marks]
Question 2 (14 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.

a) Construct a 95% confidence interval for the true lifetime of the screens.
[3 marks]

b) Is there any reason to challenge the claim of the manufacturer?[2 marks]

c) 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?[2 marks]

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.

a) Is it likely that the population distribution for the length of time at current address is normal? Explain your answer. [2 marks]

b) Given the answer above, can a 97% confidence interval be determined for the population mean? Explain why it can or cannot be determined. [2 marks]

c) Assume now that the confidence interval can validly be determined. Construct the 97% confidence interval for the population mean. [3 marks]

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Question 3 (10 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 QA Assignment 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 with a known standard deviation of $ 24,000.

a) Specify the most appropriate choice of null and alternative hypotheses to test whether there has been a decline in mean weekly sales. [2 marks]

b) Provide the formula for the test statistic and compute its value. [2 marks]

c) What is the critical value of the test using a significance level of 2.5 per cent?
[1 mark]

d) What is your statistical decision regarding the pair of hypotheses? [1 mark]

e) What is your conclusion (in non-statistical language) about mean weekly sales?[1 mark]

f) Repeat parts (c), (d) and (e) using a significance level of 1 per cent. [3 marks]

Question 4 (38 marks)

PART A

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: QA 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.]

We are interested in analysing the relationship between MCG match attendance and the combined membership of the two clubs playing in that match.

a) Produce a scatter plot of MCG attendance (dependent variable) against combined club membership (independent variable) and comment on the apparent relationship between these variables. [3 marks]

b) Calculate the correlation coefficient between MCG attendance andcombined club membership and interpret the result.[2 marks]
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Use Excel (or other software) to estimate the linear regression model:

c) Write out the estimated regression equation. [2 marks]

d) Interpret the both the estimated intercept and estimated slope. [2 marks]

e) Is combined club membership a statistically significant predictor of match attendance at the MCG? Use a significance level of 5%. [2 marks]

f) Comment on the goodness of fit of the estimated model. [2 marks]

g) Predict the attendance at a match between Adelaide (25,000 members) and Western Bulldogs (16,500 members).[2 marks]

h) How do you think the above model could be improved in order to get a better prediction of the attendance at an AFL match held at the MCG? [2 marks]

PART B

Measurements of the mean maximum annual temperature for Sydney over the period 1859-2012 are contained in the Excel file: QA Assignment 2.xls. [Data sourced from the Australian Bureau of Meteorology]

a) Produce a graph of the Sydney temperature time series. Describe the relationship between temperature and time. [3 marks]

b) Use Excel to produce a simple linear trend regression model of Sydney temperatures. Write out the estimated regression equation. [2 marks]

c) Interpret the estimated slope. [2 marks]

d) Is there a statistically significant trend in Sydney temperatures? Explain why or why not? [2 marks]

e) Comment on the model’s goodness of fit. [2 marks]

f) Use your estimated model to forecast the mean annual Sydney temperature for 2015. [2 marks]

g) Use your estimated model to forecast the mean annual Sydney temperature for 2030. [2 marks]

h) Which of the above two forecasts do you expect to be more reliable?
Explain why. [3 marks]

i) Does your analysis in this question prove that man-made environmental change is causing global warming? [3 marks]

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