Part A-
1. The following are the durations in minutes of a sample of long-distance phone calls made within Australia reported by one long-distance carrier.
|
Time (in Minutes)
|
Relative frequency
|
|
0 to < 5
|
0.36
|
|
5 to < 10
|
0.23
|
|
10 to < 15
|
0.15
|
|
15 to < 20
|
0.1
|
|
20 to < 25
|
0.08
|
|
25 to < 30
|
0.06
|
|
≥ 30
|
0.02
|
What is the cumulative relative frequency for the percentage of calls that lasted less than 15 minutes?
A. 0.84
B. 0.10
C. 0.59
D. 0.74
2. Which of the following is a survey error?
A. Coverage error
B. Measurement error
C. Sampling error
D. All of the above
3. In linear regression, the standard error of estimate (SYX) measures:
A. the variation around the regression line
B. the explained variation.
C. the variation in the X variable.
D. the variation in the Y variable.
4. The least squares regression line minimises the sum of
A. Differences.
B. Squared differences.
C. Predictions.
D. Errors.
5. The coefficient of multiple determination is always in the range of
A. Zero to 1
B. -1 to 1
C. -1 to zero
D. -0.99 to 0.99
6. In a Chi square test with 4 rows and 6 columns, the degrees of freedom for the test statistic is
A. 8
B. 4
C. 15
D. 10
7. A random sample of 100 people shows that 15 are left handed. A 95% confidence interval for the true proportion (π) of left-handers is
A. 0.05 ≤ π ≤ 0.15
B. 0.15 ≤ π ≤ 0.33
C. 0.08 ≤ π ≤ 0.22
D. 0.15 ≤ π ≤ 0.85
8. The Central Limit Theorem (CLT) is important in statistics because
A. For any sample size, the CLT says the sampling of the sample mean is approximately normal.
B. For any n, the CLT says the population is approximately normal.
C. For any population, the CLT says the sampling distribution of the sample mean is approximately normal, regardless of sample size.
D. For a large n, the CLT says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
9. The salaries of Australian football players average $150,000 with a standard deviation of $75,000. Suppose a sample of 100 football players was taken. What is the standard error for the sample mean?
A. $750,000
B. $7,500
C. $150,000
D. $7 million
10. A study was conducted to determine whether the use of seat belts in motor vehicles depended on ethnic background. A sample of 792 children treated for injuries sustained from vehicle accidents was obtained, and each child classified according to (1) ethnic background (English Speaking background (ESB) or Non English Speaking background (NESB), and (2) seat belt usage (worn or not worn) during the accident. The number of children in each category is shown below.
|
|
NESB
|
ESB
|
|
Seat belts worn
|
31
|
148
|
|
Seat belts not worn
|
283
|
330
|
Which test would you use to properly analyse the data in this study?
A. χ2 test for difference among more than two proportions.
B. χ2 test for variance.
C. χ2 test for independence.
D. None of the above.
11. A university mailed a survey to a total of 400 students. The sample included 100 students randomly selected from first year, second year, third year, and postgraduate level studying on campus last term. What sampling method was used?
A. Systematic sample
B. Stratified sample
C. Cluster sample
D. Simple random sample
12. A Type II error is committed when
A. we reject a null hypothesis that is false.
B. we do not reject a null hypothesis that is false.
C. we do not reject a null hypothesis that is true.
D. we reject a null hypothesis that is true
13. The method of least squares is used on time-series data for
A. Obtaining the trend equation.
B. Eliminating irregular movements.
C. Exponentially smoothing a series.
D. Deseasonalising the data.
14. The repeating swings over more than one year represent which component of a time series?
A. cyclical
B. seasonal
C. trend
D. irregular
15. Which of the following statements about moving averages is true?
A. All of the statements below
B. It is simpler than the method of exponential smoothing.
C. It can be used to smooth a series.
D. It gives equal weight to all values in the computation.
Part B-
Question 1-
a. A random sample of 256 Australian executives was asked how many hours per week they spent on work brought home from the office. The average was found to be 15.6 hours. Assuming a population standard deviation of 4 hours, estimate the 99% confidence interval for the mean of all Australian executives.
b. If X- = 135, S = 6.5, n = 141, and assuming that the population is normally distributed, construct a 90% confidence interval estimate of the population mean μ.
A student advocacy organisation wants to estimate the mean monthly public transport costs for students living in Perth. Based on studies conducted in other cities, the standard deviation is assumed to be $65. The group wants to estimate the mean bill within ± $10 with 99% confidence.
c. What sample size is needed?
d. If 95% confidence is desired, what sample size is needed?
Question 2- It is claimed in the newspapers that teenagers aged 16-19 years log on to social networking sites on average 10 times a week. A researcher believes that teenagers log on to social networking sites more than this and she wishes to find statistical evidence to support her belief. Let μ represent the population mean number of times 16-19 year old teenagers log on in this city to social network sites.
a. State the null and alternative hypotheses for the researcher's test.
b. Explain in the context of this scenario the meaning of Type I and Type II errors.
c. Suppose that the researcher carries out some research in this city. Based on past studies, she assumes that the standard deviation of the number of times that teenagers aged 16-19 log on to social network sites is 1.6. From a randomly selected sample of 100 teenagers aged 16-19, the researcher finds that the mean number of times that they log on per week is 10.87. At the 0.01 level of significance, is there evidence that the mean number of log-ons is greater than 10 per week?
d. What is your answer in (c) if the standard deviation is 4.0 (use a significance level of 0.01)?
e. Based on the sample data described in part (c), calculate a 95% confidence interval for the mean number of teenager log-ons assuming an estimated sample standard deviation (S) of 2.0.
Question 3- Apar Publishing Ltd is interested in understanding the relationship between the number of years that its sales representatives have worked for the company and the annual value of their sales. Data from a random sample of 12 sales representatives is shown below.
|
Number of years working at Apar Publishing (X)
|
Annual sales ($'000) for each sales representative (Y)
|
|
3
|
487
|
|
5
|
445
|
|
2
|
272
|
|
8
|
641
|
|
2
|
187
|
|
6
|
440
|
|
7
|
346
|
|
1
|
238
|
|
4
|
312
|
|
2
|
269
|
|
9
|
655
|
|
6
|
563
|
|
Mean years working at Apar 4.58
|
Mean annual sales ($'000) $404.58
|
Linear regression analysis has been undertaken with Excel and is shown in the following tables.
|
SUMMARY OUTPUT
|
|
Regression Statistics
|
|
Multiple R
|
0.8325
|
|
R Square
|
0.6931
|
|
Adjusted R Square
|
0.6624
|
|
Standard Error
|
92.1055
|
|
Observations
|
12
|
|
ANOVA
|
|
df
|
SS
|
MS
|
F
|
Significance F
|
|
Regression
|
1
|
191600.622
|
191600.6
|
22.58528
|
0.00078
|
|
Residual
|
10
|
84834.29469
|
8483.429
|
|
|
|
Total
|
11
|
276434.9167
|
|
|
|
|
Coefficients
|
Standard Error
|
t Stat
|
P-value
|
Lower 95%
|
Upper 95%
|
|
Intercept
|
175.8288
|
54.9899
|
3.1975
|
0.0095
|
53.3037
|
298.3539
|
|
X (Years)
|
49.9101
|
10.5021
|
4.7524
|
0.0008
|
26.5100
|
73.3102
|
a. State the linear regression equation for this data.
b. Interpret the meaning of the slope coefficient, b1 in this problem.
c. Predict the value of annual sales when the sales representative has worked (i) 3.5 years, and (ii) 12 years. Explain which of these two predictions (i.e., (i) or (ii)) is likely to be more reliable and why this may be the case.
d. Comment on the goodness of fit of the estimated regression model.
e. At the 0.01 level of significance, is there evidence of a linear relationship between years working at Apar Publishing and the value of annual sales?
Question 4- An analysis of economic development strategies suggests that some Asian economies benefit from having policies that promotes growth in manufacturing and growth of exports. A regression analysis has been undertaken with data from a sample of countries that operate these types of policies.
|
Country
|
Growth/Annum GDP
|
Growth/Annum manufacturing X1
|
Growth/Annum exports X2
|
|
China
|
11.0
|
7.3
|
14.4
|
|
Hong Kong
|
7.1
|
10.2
|
6.2
|
|
Japan
|
10.5
|
6.0
|
14.0
|
|
Korea
|
9.7
|
13.1
|
13.8
|
|
Malaysia
|
4.9
|
8.0
|
9.8
|
|
Singapore
|
6.1
|
5.9
|
8.1
|
|
Thailand
|
7.0
|
8.1
|
12.8
|
|
|
Mean growth/annum GDP = 8.04
|
Mean X1 = 8.37
|
Mean X2 = 11.3
|
A summary of the multiple regression analysis undertaken in Microsoft Excel is shown below.
|
SUMMARY OUTPUT
|
|
Regression Statistics
|
|
Multiple R
|
0.746
|
|
R Square
|
0.556
|
|
Adjusted R Square
|
0.334
|
|
Standard Error
|
1.918
|
|
Observations
|
7
|
|
ANOVA
|
|
df
|
SS
|
MS
|
F
|
Significance F
|
|
Regression
|
2
|
18.439
|
9.220
|
2.506
|
0.197
|
|
Residual
|
4
|
14.718
|
3.679
|
|
|
|
Total
|
6
|
33.157
|
|
|
|
|
Coefficients
|
Standard Error
|
t Stat
|
P-value
|
Lower 95%
|
Upper 95%
|
|
Intercept
|
1.389
|
3.781
|
0.367
|
0.732
|
-9.109
|
11.886
|
|
Growth/Annum manufacturing X1
|
0.075
|
0.308
|
0.245
|
0.819
|
-0.780
|
0.931
|
|
Growth/Annum exports X2
|
0.533
|
0.240
|
2.221
|
0.091
|
-0.133
|
1.199
|
a. State the multiple regression equation for this data.
b. Interpret the meaning of the slope coefficients in this problem.
c. Predict the GDP growth for a country that has adopted these manufacturing (X1) and export (X2) supportive policies with manufacturing growth of 9.5 and export growth of 10.7.
d. Comment on the goodness of fit of the estimated regression model.
e. Is there a significant relationship between GDP growth and the two independent variables (i.e., policies supporting growth in manufacturing and policies supporting growth in exports) at the 0.05 level of significance?
f. Explain how you think the above model could be improved?
Question 5- Non-response to mail, telephone and interview surveys is a continual problem for researchers. The problem knows whether the individuals who do not respond form a separate subsample of the population. A common way of encouraging survey response is to offer a prize draw. To test whether offering a prize is effective in increasing the response rate, two groups of financial planners are surveyed by mail, one in Melbourne without offering a prize, and the other in Sydney where a prize draw for survey participants was offered. The results are provided in the following table:
|
City
|
Melbourne
|
Sydney
|
Total
|
|
Response
|
85
|
102
|
187
|
|
Non-response
|
142
|
118
|
260
|
|
Total
|
227
|
220
|
447
|
Is there evidence of a significant difference between the two treatment groups in the proportion of responses to the survey? Please answer this question using a chi-square test for differences between proportions. You should assume a significance level of 0.05 and answer this question using the following steps:
a. State the hypotheses that you are testing
b. Determine the critical value of χ2 and your decision rule
c. Calculate the χ2 test statistic for this data
d. State your conclusion.
Question 6- Consider a 5-year moving average used to smooth a time series that was first recorded in the year 1955.
a. Which year serves as the first centred value in the smoothed series?
b. How many years of values in the series are lost when calculating all the 5-year moving averages?
You are using exponential smoothing on an annual time series concerning total revenues (in millions of constant 1995 dollars). If you are using a smoothing coefficient of ω = 0.20 and the exponentially smoothed value for 2006 is E2006 = (0.20)(12.1) + (0.80)(9.4):
c. What is the smoothed value of this series in 2006?
d. What is the smoothed value of this series in 2007 if the value of the series in that year is 11.5 million of constant 1995 dollars?
The table below shows the seasonal index for the occupancy rate of a hotel.
|
Seasonal Index for Hotel Occupancy Rate
|
|
Season
|
Index
|
|
Quarter 1
|
0.878
|
|
Quarter 2
|
1.076
|
|
Quarter 3
|
1.171
|
|
Quarter 4
|
0.875
|
e. Using the seasonal index, show for each quarter how much expected hotel occupancy is likely to vary (+/-) from the mean annual occupancy in percentage terms.