1137 Mathematical Applications Assessment Answer

pages Pages: 4word Words: 890

Question :

Task 2         

Mathematics

Course title and codeMathematical Applications,  Course Code 1137(Tertiary)
Semester unit title, codes and valueUnit 3 Mathematical Applications, Unit Code 11306
1.0 for Semester Unit 
Assessment item typeAssignment
Marked out ofA synthetic score will be generated from z-scores

 Year 12 Mathematical Applications assignment

Your assignment is on bivariate data analysis. You will do two investigations into the relationships of two different pairs of numerical variables, the first using a two-way table and the second using your knowledge of linear regression. You will choose the two pairs of numerical variables. You will demonstrate your knowledge of the mathematical ways of examining data and present your findings. The assignment is in three parts, your data selection, a two-way table analysis and a scatter plot analysis.

You are expected to include all graphs and table in one Word document after creating them in Excel or Google sheets.

Part 1

You will find the data for your scatter plot analysis and submit it to your classroom teacher. This is to ensure that you are on the right track and have suitable data that is in the right form for you to begin your analysis. This is due Thursday of week 13 by 10 pm.

Your submission should be a google sheet or excel spreadsheet with your data in table form. You should have two sets of data that you are going to compare.

Part 2 - 30%

You will select two questions from the questionnaire and use a two-way table to explore the relationship between them. Note that there may be no relationship—this is fine. You will need to discuss what about the data made you conclude that there was no relationship. Then you must discuss your data and what the tables show you about the data. You should write 200 words on this, including answering questions about the data that you pose yourself—for example, ‘What percentage of people who like A dislike B?’

For this part of the assignment, you must also include the following clearly labelled tables:

1.) The numerical two-way table;

2.) Percentage two-way tables by rows and by columns;

3.) The total percentage two-way table.

Part 3 - 70%

For this part of the assignment, you need to find two pairs of numerical variables to form the basis of your investigation.

Here is a short list of some possible examples of pairs of numerical variables that you could investigate:

•height and intelligence;

•temperature and rainfall for a particular location or region;

•latitude and skin cancer deaths;

•BMI (body mass index) and time in surgery;

•heights of husbands and heights of wives at marriage;

•wealth and happiness;

•years married and life expectancy;

•GDP (gross domestic product) per capita and life expectancy;

Note that these are only suggestions—the possibilities are immense.

Show More

Answer :

Part 2 Answer

For this purpose, two questions selected are:

  1. Do you play cricket?
  2. Do you own a bicycle?

The numerical two-way table is as follows:

COUNTDo you play cricket?Do you own a bicycle?TOTALS
Yes8112120
No15248200
TOTALS160160320

It can be seen that out of a sample set of 160, 8 play cricket while remaining 152 do not play cricket. It can be seen that out of a sample set of 160, 112 own a bicycle while remaining 48 do not own a bicycle.

Hence, overall there are 120 positive responses and 200 negative responses.

It was also seen that out of 8 who play cricket, 7 also own bicycle. While out of 152 those who don’t play cricket, 105 own bicycle. Hence, it seems there is no relation between two variables and bicycle is more popular than playing cricket.

Percentage two-way tables by rows and by columns and the total percentage two-way table are as follows:

PERCENTAGESDo you play cricket?Do you own a bicycle?TOTALS
Yes5.0%70.0%37.5%
No95.0%30.0%62.5%
TOTALS100.0%100.0%100.0%

Instead of numerical count, the above table presents same data in form of percentages which is easier to compare across categories, especially when the sample size is different. We can see that only 5% of respondents play cricket with an overwhelming 95% not playing cricket. The response is very different when it comes to owning bicycles. As many as 70% of the respondents own bicycle with only 30% who do not own a bicycle. 

Due to majority negative responses for cricket question, the overall negative responses account for 62.5% with only 37.5% positive responses.

Part 3 Answer

Using data provided by WorldBank, Australian GDP and lending interest rate data have been taken for 40 years ranging from 1979 till 2018:

 GDP (current US$)GDP (current US$ bn)Lending interest rate (%)
1979     1347120,29,989.78                           135                     9.13 
1980     1497749,30,362.12                           150                     9.98 
1981     1766422,84,918.15                           177                   11.83 
1982     1937702,74,743.46                           194                   13.29 
1983     1770304,16,471.69                           177                   12.42 
1984     1932421,66,274.23                           193                   11.50 
1985     1802347,16,575.90                           180                   12.42 
1986     1820369,33,407.95                           182                   15.00 
1987     1890603,49,391.21                           189                   15.08 
1988     2356591,96,740.40                           236                   14.08 
1989     2992679,74,920.61                           299                   16.46 
1990     3107772,22,008.47                           311                   16.35 
1991     3253104,15,195.04                           325                   13.42 
1992     3248788,74,105.98                           325                   10.58 
1993     3115444,06,970.21                           312                     9.42 
1994     3222116,91,456.24                           322                     9.09 
1995     3672163,64,716.37                           367                   10.50 
1996     4003027,31,411.23                           400                     9.73 
1997     4345680,07,512.91                           435                     7.17 
1998     3988991,38,574.24                           399                     6.68 
1999     3886082,21,581.65                           389                     6.57 
2000     4152226,33,925.77                           415                     7.72 
2001     3783760,86,723.19                           378                     6.84 
2002     3946489,11,678.53                           395                     6.36 
2003     4664880,60,570.76                           466                     6.61 
2004     6124903,96,927.02                           612                     7.05 
2005     6934077,58,231.85                           693                     7.26 
2006     7460542,07,846.66                           746                     7.61 
2007     8530996,30,996.31                           853                     8.20 
2008    10539955,23,724.26                        1,054                     8.91 
2009     9278051,83,330.88                           928                     6.02 
2010    11461384,65,603.81                        1,146                     7.28 
2011    13966499,06,339.35                        1,397                     7.74 
2012    15461517,83,872.96                        1,546                     6.98 
2013    15761844,67,015.49                        1,576                     6.18 
2014    14674837,05,131.74                        1,467                     5.95 
2015    13516939,84,524.50                        1,352                     5.58 
2016    12088469,93,739.99                        1,209                     5.42 
2017    13301357,56,844.41                        1,330                     5.25 
2018    14339043,48,500.12                        1,434                     5.26 

The hypothesis is to test whether there is a negative relationship between the GDP and lending interest rate. In other words, as GDP increases, lending rate decreases and vice versa. 

Scatter plot is as follows:Scatter plot

Regression was done using excel:Regression

The R is 0.66 which indicates reasonable positive relationship between the variables as r ranges from -1 to 1. P-value is less than significance value of 0.05 indicating that the model and coefficients are statistically significant. The regression equation is y = 1,489.75 - 94.6x where y is predicted GDP in USDbn and x is the lending interest rate.

The residuals data and plot are:

ObservationPredicted GDP (current US$ bn)ResidualsStandard Residuals
1                                          626.57              -491.86                         -1.38 
2                                          545.77              -396.00                         -1.11 
3                                          370.38              -193.73                         -0.54 
4                                          232.42                -38.65                         -0.11 
5                                          315.20              -138.16                         -0.39 
6                                          401.91              -208.67                         -0.59 
7                                          315.20              -134.96                         -0.38 
8                                            70.82               111.21                          0.31 
9                                            62.94               126.12                          0.35 
10                                          157.54                 78.12                          0.22 
11                                           -67.13               366.40                          1.03 
12                                           -57.27               368.05                          1.03 
13                                          220.60               104.71                          0.29 
14                                          488.62              -163.74                         -0.46 
15                                          598.98              -287.44                         -0.81 
16                                          630.12              -307.91                         -0.87 
17                                          496.50              -129.29                         -0.36 
18                                          569.42              -169.12                         -0.48 
19                                          811.82              -377.25                         -1.06 
20                                          857.54              -458.64                         -1.29 
21                                          868.58              -479.97                         -1.35 
22                                          759.79              -344.57                         -0.97 
23                                          842.56              -464.19                         -1.30 
24                                          887.89              -493.24                         -1.39 
25                                          864.24              -397.75                         -1.12 
26                                          822.86              -210.37                         -0.59 
27                                          803.15              -109.74                         -0.31 
28                                          769.65                -23.59                         -0.07 
29                                          714.47               138.63                          0.39 
30                                          647.07               406.93                          1.14 
31                                          920.21                  7.59                          0.02 
32                                          801.18               344.96                          0.97 
33                                          757.82               638.83                          1.79 
34                                          829.95               716.20                          2.01 
35                                          905.23               670.95                          1.89 
36                                          926.91               540.57                          1.52 
37                                          962.38               389.31                          1.09 
38                                          976.97               231.88                          0.65 
39                                          993.55               336.58                          0.95 
40                                          992.10               441.80                          1.24 

lending interest rate graph residual plotLending interest rate: Line fit plot graph


The above residual plot indicates a definite pattern where residuals move up and down in a wave like pattern. Hence, the residuals may indicate that the regression model may not be linear but a second order model.

Conclusion

From above scatter plot and regression, we can conclude that there is definitely a relationship between GDP and lending interest rates in Australia. However, the relationship is positive and also regression equation that should be used is most probably of second order rather than linear as indicated by residual plot.