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CSEC Additional Mathematics SBA

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An Investigation of LIME and Digicel Mobile Coverage in Vide-Bouteille

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CSEC Additional Mathematics SBA

  1. 1. St. Mary’s College An Investigation of LIME and Digicel Mobile Coverage in Vide-Bouteille Ad Math SBA - Statistics Ronaldo Degazon
  2. 2. Contents Title...................................................................................................................................................1 Introduction.......................................................................................................................................2 Purpose of the Project ........................................................................................................................3 Variables............................................................................................................................................4 Data Collection...................................................................................................................................5 Data Collected for LIME......................................................................................................................6 Data Collected for Digicel..................................................................................................................13 Hypothesis Test...........................................................................................................................19-22 Mathematical Knowledge ............................................................................................................ 23-25 Analysis...................................................................................................................................... 26- 27 Findings...........................................................................................................................................28 Recommendations............................................................................................................................29 Conclusion .......................................................................................................................................30 Appendix & Bibliography...................................................................................................................31 Bibliography.....................................................................................................................................33
  3. 3. Page | 1 Title An investigation of LIME and Digicel mobile signal strength in the area of Vide-Bouteille. Vs
  4. 4. Page | 2 Introduction Cell phone signal is a duplex communication system, which means that the two devices involved both send and receive information. The first device involved is the cellular phone and the second device is the cell site. To communicate with each other, a cell site sends radio waves to a cellular phone and in turn, the cell phone sends radio waves to the cell site. This method of communication is what allows cellular phones to get access to a network. A mobile phone signal (or reception) is the signal strength of the connection to the mobile phone with its network. Both cell phones and cell sites have use transmitters to send the radio signals and use receivers to receive the radio signals. The transmitters and receivers on the cell sites however, are much more powerful than those on a cell phone, thus mobile reception is much more dependent upon the cell site than the cell phone. Though phones use a set of 5 bars to represent signal strength it is formally measured in dBm (short for dBmW). This is a power ratio of deciBels per milliWatt. It is measured on a scale starting from 0dBm which is the strongest possible signal to -113dBm. However, because cell phones are not as strong as the cell sites, a signal of 0dBmis technically impossible. Excellent signal strength is usually above -60 dBm whilst poor signal strength is less than -90 dBm. When an area is unable to receive sufficient signal strength it is called a dead zone. This usually occurs at signal strengths less than -113 dBm. Cell sites emit radio waves at different frequencies. These different frequencies are classified into frequency bands. The four frequency bands that cells sites and cell phones operate on in St. Lucia are 850MHz, 900MHz, 1800 MHz and 1900 MHz. Some cell sites operate on only one frequency whilst others may operate on two frequencies. Those cell sites which operate on two frequency bands are describes as dualband. The lower the frequency is, the more powerfully that the radio waves will be emitted. So of the four frequency bands mentioned, an 850MHz would provide the most powerful signal. However, as you will see later in this presentation there are many variables which can greatly affect signal strength. In St. Lucia, there is a duopoly of cellular network providers – LIME and Digicel. LIME cell sites operate on 850MHz and 1900 MHz bands, whilst Digicel cell sites operate on 900MHz, 1800MHz and 1900 MHz bands.
  5. 5. Page | 3 Purpose of the Project LIME and Digicel are the sole cellular service providers in St. Lucia. In my area of residence, Vide-Bouteille, Castries, I receive excellent reception on my LIME cell phone, but the question still holds, would I receive the same coverage with Digicel? This project is aimed at discovering the following objectives: οƒΌ What is the average mobile reception for LIME phones in Vide-Bouteille? οƒΌ How consistent is mobile reception for LIME phones in Vide-Bouteille? οƒΌ What is the mean mobile reception for Digicel phones in Vide-Bouteille? οƒΌ How consistent is mobile reception for Digicel phones in Vide-Bouteille? οƒΌ Which provider gives better coverage in Vide-Bouteille or is their coverage the same? (This objective will be answered using a hypothesis test.) Provided that it is discovered that one mobile company actually provides better coverage in the area, residents of Vide-Bouteille can be better advised on which mobile service provider they should choose.
  6. 6. Page | 4 Variables 1. Distance/Line of Sight from Cell Site The further the line of sight away from a cell site, the weaker the reception will be and vice-versa. 2. Frequency of Cell Site The lower the frequency is, the more powerful signals that a cell site can emit. 3. Phone Different phones have receivers of different strengths. It was ensured that two Blackberries of the same model (9790) were used to eliminate variation of this factor. 4. Building Structure The materials with which obstructing buildings are built affect cell reception. Out of plywood, stone, concrete and steel, plywood obstructs radio signals the least. 5. Network Traffic The number of cell phones communicating with the tower affects signal strength. The more traffic, the weaker reception may be. 6. Terrain & Vegetation It is extremely difficult to provide cellular coverage in a heavily forested area. Also if massive hills are in the way of radio signals then cell reception is significantly weakened. 7. Weather Precipitation such as rain, sleet, snow, hail etc. slightly weaken mobile signal strength. It would be extremely difficult to variables 1, 4, and 6 constant at the same time. This is due to the fact that the cells sites for each provider are located in different areas. If the data is collected using constant distances from cell sites, the terrain, vegetation and buildings which obstruct the radio signals will differ. If however, variables 4 and 6 are kept constant by using the same locations to measure reception, the distances from the cell sites would differ. Furthermore, network traffic for each provider’s cell site will differ and cannot be controlled. Due to all these restrictions concerning the control of variables, areas were chosen randomly where the reception for each provider will be tested.
  7. 7. Page | 5 Data Collection Due to the difficulty of holding the desired variables constant (as mentioned in the variables section), a random selection was used. Fifteen random areas in Vide-Bouteille were chosen and in each area the signal strength was recorded with using two separate phone – 1 LIME Blackberry and 1 Digicel Blackberry. Due to the fact that signal strength is constantly fluctuates, ten readings were taken for each area and the mean was used as the estimate for the signal strength in that area. Below is the data collection process: 1. Ten readings were taken at 10 seconds intervals for Area 1 using the LIME Blackberry. 2. The Cell ID was also noted. 3. The mean of the readings was calculated. 4. Steps 1-3 were repeated using the Digicel Blackberry. 5. Steps 1-4 were repeated for Areas 2-15. The results were then recorded in a table. The Cell ID was used to locate the tower that the signal reception was coming from. Then, using Google Earth, the distance of each cell tower from the location was found. 6. The ground level (horizontal) distance from the two points was found. 7. The elevation (vertical distance) of the cell tower from the point was found. 8. Using basic trigonometric ratios, the line of sight distance of the point from the cell tower was calculated.
  8. 8. Data Collected for LIME
  9. 9. Page | 6 Raw Data Collected for LIME1 Taken on Saturday 2nd February 2013: 2:45 - 4:30 pm – Weather: Partly Cloudy Area # Readings 1 2 3 4 5 6 7 8 9 10 Mean Cell ID 1 -63 -64 -65 -64 -69 -63 -66 -65 -64 -69 -65.2 22573 2 -60 -66 -64 -62 -65 -73 -64 -63 -63 -63 -64.3 22576 3 -65 -69 -67 -64 -61 -67 -65 -64 -65 -62 -64.9 22661 4 -82 -77 -72 -72 -72 -72 -72 -69 -66 -65 -71.9 22644 5 -56 -49 -47 -47 -48 -46 -49 -47 -46 -46 -48.1 22575 6 -80 -92 -82 -80 -75 -76 -76 -77 -81 -88 -80.7 22662 7 -74 -75 -72 -71 -74 -74 -78 -78 -78 -78 -75.2 22524 8 -86 -91 -91 -82 -86 -83 -84 -83 -82 -81 -84.9 22641 & 22644[1] 9 -88 -85 -85 -82 -85 -84 -85 -88 -86 -81 -84.9 22576 10 -67 -63 -74 -69 -66 -78 -65 -62 -63 -68 -67.5 22521 & 22524[1] 11 -87 -81 -89 -91 -88 -90 -87 -81 -86 -84 -86.4 22521 12 -93 -80 -79 -79 -79 -79 -80 -80 -81 -83 -81.3 22644 13 -103 -108 -110 - - - - - - - -107 22644 14 -84 -97 -92 -93 -91 -89 -89 -64 -63 -64 -91 22644 15 -81 -74 -80 -77 -75 -71 -74 -75 -78 -73 -75.8 22576 Below: Table showing results recorded with LIME Blackberry The table above shows the results recorded with the LIME Blackberry. Based on the raw data, it appears that most LIME customers receive average to good cell reception. Area #5 had excellent cell reception which indicates that it must have been in within extremely near proximity to a cell site. However area 13 had astoundingly shocking reception. Out of the 10 readings, 7 times the phone received no reception! [1] These Cell ID’s arefrom the same cell site,but a different frequency band.
  10. 10. Page | 7 Mean Signal Strength / dBm Frequency -110 - -101 1 -100 - -91 1 -90 - -81 5 -80 - -71 3 -70 - -61 4 -60 - -51 0 -50 - -41 1 Above: Frequency Table for Signal Strength Class Intervals Above: Histogram Showing Frequencies of Signal Strength Class Intervals The range -90dBm - -81dB yielded the highest frequency of 5. The second highest frequency of 4 fell in the range -70dBm - -61dBm whilst the third highest frequency of 3 fell in the range -80dBm - -71dBm. Three different ranges yielded a frequency of 1 whilst the range -60dBm - - 51dBm yielded no readings. The majority of the readings are in the center of the distribution, meaning that the readings almost follow a normal distribution. There is no distinct positive or negative skewness. 0 1 2 3 4 5 6 -110 - -101 -100 - -91 -90 - -81 -80 - -71 -70 - -61 -60 - -51 -50 - -41 Frequency Mean Signal Strength/dBm Frequency Histogram for LIME Readings in Vide- Bouteille
  11. 11. Page | 8 Mean Signal Strength / dBm Cumulative Frequency < -100.5 1 < -90.5 2 < -80.5 7 < -70.5 10 < -60.5 14 <-50.5 14 < -40.5 15 Above: Cumulative Frequency Curve for LIME Signal Strength Readings 0 2 4 6 8 10 12 14 16 -100.5 -90.5 -80.5 -70.5 -60.5 -50.5 -40.5 CumulativeFrequency Class Cumulative Frequency Curve for LIME Readings in Vide-Bouteille
  12. 12. Page | 9 Above: Box & Whiskers Plot Illustrating LIME Data The above Box & Whiskers plot gives an idea of how the LIME readings are distributed. The highest reading was -48.1 dB whilst the lowest reading was -107dB. The inter-quartile range represented by the box, shows the range of the middle 50% of the distribution and was calculated as 19.7dB. Looking even closer, the upper semi-inter-quartile range (light-grey shaded) was calculated as 10.6 dB whilst the lower semi-inter-quartile range (dark-grey shaded) was calculated as 9.1 dB. The fact that these ranges are almost equal indicates that the distribution of the LIME readings is almost symmetrical. The calculation of the figures is shown below: - Median = 1 2 ( 𝑛 + 1) π‘‘β„Ž π‘£π‘Žπ‘™π‘’π‘’ = 8th Highest value ------------> -75.8 dB - Lower Quartile (Q1) = -84.9 dB - Upper Quartile (Q3) = -65.2 dB - Inter-quartile Range = Q3 – Q1 --------> -65.2 – (-84.9)dB -----------> 19.7dB - Upper Semi-Inter-Quartile Range = Q3 – Q2 ---------> -65.2 – (-75.8)dB --------> 10.6dB - Lower Semi-Inter-quartile Range = Q2 – Q1 ---------> -75.8 – (-84.9)dB --------> 9.1dB -110 -100 -90 -80 -70 -60 -50 -40 Readings LIME Box and Whisker Plot for LIME Readings in Vide- Bouteille
  13. 13. Page | 10 Calculations Sample Mean of Cell Readings This is found by taking the sum of the sample readings and dividing them by the number of samples. Below is the formula: 𝑋̅ = Ξ£X n Where 𝑿̅ = sample mean πšΊπ‘Ώ = sum of values n = number of values (-65.2) + (-64.3) + (-64.9) + (-71.9) + (-48.1) + (-80.7) + (-75.2) + (-84.9) + (-84.9) + (-67.5) + (-86.4) + (-81.3) + (-107) + (-91) + (-75.8) 15 𝑿̅= (-1149.1) dB / 15 𝑿̅ = -76.6 dB Variance The variance is found by subtracting each value from the mean, squaring the value, then dividing the sum of the squares by the number of values 𝜎2 = Ξ£( 𝑋 βˆ’ 𝑋̅)2 𝑛 βˆ’ 1 Where 𝝈 𝟐 = Variance 𝑿 = Sample Value 𝑿̅ = Sample Mean n = number of samples = [(-65.2) – (76.6)]2 + [(-64.3) - (76.6)]2 + [(-64.9) – (76.6)]2 + [(-71.9) - (76.6)]2 + [(-48.1) - (76.6)]2 + [(-80.7) - (76.6)]2 + [(-75.2) - (76.6)]2 + [(-84.9) - (76.6)]2 + [(-84.9) - (76.6)]2 + [(-67.5) - (76.6)]2 + [(-86.4) - (76.6)]2 + [(-81.3) - (76.6)]2 + [(-107) - (76.6)]2 + [(-91) - (76.6)]2 + [(-75.8) - (76.6)]2 15 - 1 𝝈 𝟐 = 195.87 dB
  14. 14. Page | 11 Standard Deviation This represents the average distance of a value from the mean. It is simply found by taking the square root of the variance: 𝜎 = √ Ξ£( 𝑋 βˆ’ 𝑋̅)2 𝑛 βˆ’ 1 Where 𝝈 = standard deviation 𝝈 𝟐 = Variance 𝑿 = Sample Value 𝑿̅ = Sample Mean n = number of samples 𝜎 = √195.87 𝜎 = 14 dB
  15. 15. Data Collected for Digicel
  16. 16. Page | 13 Raw Data Collected for Digicel Taken on Saturday 2nd February 2013: 2:45 - 4:30 pm – Weather: Partly Cloudy Area # Readings 1 2 3 4 5 6 7 8 9 10 Mean Cell ID 1 -41 -50 -49 -52 -52 -51 -46 -47 -46 -43 -47.7 61093 2 -51 -57 -54 -66 -58 -56 -60 -56 -65 -56 -57.9 61093 3 -64 -64 -63 -62 -61 -59 -59 -60 -61 -61 -61.4 61093 4 -66 -66 -67 -67 -66 -67 -67 -65 -63 -64 -65.8 61102 5 -88 -77 -83 -74 -78 -78 -77 -80 -72 -72 -77.9 61092 & 61583[2] 6 -88 -90 -89 -88 -89 -88 -94 -89 -100 -94 -90.9 61093 7 -58 -58 -53 -54 -60 -53 -96 -58 -52 -54 -59.6 61431 8 -72 -76 -71 -82 -70 -69 -71 -69 -70 -70 -72 61433 9 -52 -47 -48 -49 -50 -53 -58 -56 -57 -57 -52.7 61431 10 -73 -85 -84 -80 -78 -71 -70 -67 -68 -70 -74.6 61102 & 61531[2] 11 -78 -81 -81 -83 -85 -85 -84 -80 -76 -78 -81.1 61531 12 -85 -68 -68 -70 -68 -68 -68 -70 -70 -70 -70.5 61102 13 -71 -76 -72 -73 -71 -73 -71 -65 -67 -76 -71.5 61431 14 -57 -57 -60 -60 -61 -58 -56 -58 -57 -57 -58.1 61431 15 -74 -75 -73 -72 -73 -73 -68 -70 -75 -77 -73 61093 & 61102[2] Above: Table Illustrating Readings Recorded With Digicel Blackberry The table above illustrates the raw data collected with the Digicel Blackberry. Based on this sample of data, it appears that Digicel provides good coverage throughout Vide-Bouteille. There were some areas with excellent reception which possibly indicates close proximity to a cell site, such as Area #1. However, there was a specific area which received low reception – Area #6. 2 [2] These Cell ID’s arefrom the same cell site,but a different frequency band.
  17. 17. Page | 14 Mean Signal Strength / dBm Frequency -110 - -101 0 -100 - -91 1 -90 - -81 1 -80 - -71 6 -70 - -61 2 -60 - -51 4 -50 - -41 1 Above: Frequency Table for Signal Strength Class Intervals Above: Histogram Illustrating Frequencies of Signal Strength Class Intervals The range -80dBm - -71 dBm yielded the highest frequency of 6 mean signal strength readings. The second highest frequency of 4 fell within the range -60dBm - -51dBm whilst the third highest frequency of 2 fell within the range -70dBm - -61dBm. Three ranges yielded a frequency of 1 each whilst there were no mean signal strength readings which fell withing the range -110dBm - -101dBm. 0 1 2 3 4 5 6 7 -110 - -101 -100 - -91 -90 - -81 -80 - -71 -70 - -61 -60 - -51 -50 - -41 Frequency Mean Signal Strength / dBm Frequency Histogram for Digicel Readings in Vide-Bouteille
  18. 18. Page | 15 Mean Signal Strength / dBm Cumulative Frequency < -100.5 0 < -90.5 1 < -80.5 2 < -70.5 8 < -60.5 10 <-50.5 14 < -40.5 15 Above: Cumulative Frequency Table Above: Cumulative Frequency Curve for Digicel Readings 0 2 4 6 8 10 12 14 16 -100.5 -90.5 -80.5 -70.5 -60.5 -50.5 -40.5 Frequency Class Cumulative Frequency Curve for Digicel Readings in Vide-Bouteille
  19. 19. Page | 16 Above: Box & Whiskers Plot for Digicel Readings The above Box & Whiskers plot gives an idea of how the Digicel readings are distributed. The highest reading was -47.7dB whilst the lowest reading was -90.9 dB. The inter-quartile range represented by the box, shows the range of the middle 50% of the distribution and was calculated as 14dB. Looking even closer, the upper semi-inter-quartile range (red shaded) was calculated as 10.6 dB whilst the lower semi-inter-quartile range (pinkish shaded) was calculated as 9.1 dB. This indicates a distinctly positive skew, which indicates that Digicel readings were consistently high throughout Vide-Bouteille. - Median = 1 2 ( 𝑛 + 1) π‘‘β„Ž π‘£π‘Žπ‘™π‘’π‘’ = 8th Highest value ------------> -70.5 dB - Upper Quartile = -58.8 dB - Lower Quartile = -72.8 dB - Inter-quartile Range = Q3 – Q1 --------> -58.8 – (-72.8) dB ---------> 14 dB - Upper Semi-Inter-Quartile Range = Q3 – Q2 ---------> -58.8 – (-70.5)dB --------> 11.7 dB - Lower Semi-Inter-quartile Range = Q2 – Q1 ---------> -70.5 – (-72.8)dB --------> 2.3 dB -100.5 -90.5 -80.5 -70.5 -60.5 -50.5 -40.5 Readings Digicel Box & WhiskersPlot Illustrating Digicel Readings in Vide-Bouteille
  20. 20. Page | 17 Calculations Sample Mean of Cell Readings This is found by taking the sum of the sample readings and dividing them by the number of samples. Below is the formula: 𝑋̅ = Ξ£X n Where 𝑿̅ = sample mean πšΊπ‘Ώ = sum of values n = number of values (-47.7) + (-57.9) + (-61.4) + (-65.8) + (-77.9) + (-90.9) + (-59.6) + (-72) + (-52.7) + (-74.6) +(-81.1) + (-70.5) + (-71.5) + (-58.1) + (-73) 15 𝑿̅ = -1014.7 / 15 𝑿̅ = -67.6 Variance The variance is found by subtracting each value from the mean, squaring the value, then dividing the sum of the squares by the number of values 𝜎2 = Ξ£( 𝑋 βˆ’ 𝑋̅)2 𝑛 βˆ’ 1 Where 𝝈 𝟐 = Variance 𝑿 = Sample Value 𝑿̅ = Sample Mean n = number of samples [(-47.7) – (67.6)] + [(-57.9) – (67.6)] + [(-61.4) – (67.6)] + [(-65.8) – (67.6)] + [(-77.9) – (67.6)] + [(-90.9) – (67.6)] + [(-59.6) – (67.6)] + [(-72) – (67.6)] + [(-52.7) – (67.6)] + [(-74.6) – (67.6)] + [(-81.1) – (67.6)] + [(-70.5) – (67.6)] + [(-71.5) – (67.6)] + [(-58.1) – (67.6)] + [(-73) – (67.6)] 15 - 1 𝝈 𝟐 = 131.82 dB
  21. 21. Page | 18 Standard Deviation This represents the average distance of a value from the mean. It is simply found by taking the square root of the variance: 𝜎 = √ Ξ£( 𝑋 βˆ’ 𝑋̅)2 𝑛 βˆ’ 1 Where 𝝈 = standard deviation 𝝈 𝟐 = Variance 𝑿 = Sample Value 𝑿̅ = Sample Mean n = number of samples 𝜎 = √123.03 𝜎 = 11.48 dB
  22. 22. Hypothesis Test
  23. 23. Page | 19 Hypothesis Test First glances at the measures of central tendency (mean and mode) for LIME and Digicel reception in Vide-Bouteille will draw one to conclude that Digicel’s coverage is better than LIME’s. However, the 15 areas tested are only a mere sample of the entire population of areas that can be tested. So do the samples give enough evidence to prove that LIME and Digicel coverage are not the same in Vide-Bouteille? We will use the hypothesis test to come to a conclusion. Null Hypothesis Let us assume that the mean signal strengths of LIME and Digicel in Vide-Bouteille are the same. This is our null hypothesis. H0: 𝑿1 – 𝑿2 = 0 (there is no difference between LIME and Digicel coverage in Vide-Bouteille) Where H0 – null hyptohesis 𝑿1 = mean cell reception for LIME phones 𝑿2 = mean cell reception for Digicel phones Alternative Hypothesis If however, there is enough evidence to reject our null hypothesis, we would conclude that LIME and Digicel coverage are not the same. This is our alternative hypothesis: Ha: 𝑿1 – 𝑿2 β‰  0 (there is a difference between LIME and Digicel coverage in Vide-Bouteille) Where Ha – alternative hypothesis Confidence / Significance Level This hypothesis test is set at a confidence level of 95% - we are 95% sure that that LIME and Digicel coverage are the same in Vide-Bouteille. Thus we are 5% sure that the null this is not true. 5% (0.05) is called the level of significance or the alpha level. 𝛼 = 0.05 Where 𝛼 = level of significance
  24. 24. Page | 20 T- Score We are going to need a value to test the hypothesis. This is called the t-value or t-score or test statistic. It is essentially, the number of standard deviations of the sample away from the mean. The formula for the t-value is 𝑑 = ( 𝑋̅1 βˆ’ 𝑋̅2) βˆ’ ( πœ‡1 βˆ’ πœ‡2) 𝜎 𝑋1βˆ’π‘‹2 Where 𝑿̅ 𝟏= LIME sample mean 𝑿̅ 𝟐 = Digicel Sample Mean 𝝁 𝟏= LIME population mean 𝝁 𝟐= Digicel population mean 𝝈 𝑿 πŸβˆ’π‘Ώ 𝟐 = standard error for the sampling distribution of 𝑋̅1 βˆ’ 𝑋̅2 𝜎 𝑋1 βˆ’π‘‹2 = 𝜎.Μ‚ √ 1 𝑛1 + 1 𝑛2 Where πˆΜ‚ = pooled estimate of the population standard deviation 𝒏 𝟏= number of LIME samples 𝒏 𝟐= number of Digicel samples πœŽΜ‚ = √ ( 𝑛1 βˆ’ 1) 𝑠1 2 + ( 𝑛2 βˆ’ 1) 𝑠2 2 𝑛1 + 𝑛2 βˆ’ 2 Where 𝒔 𝟏= standard deviation of LIME samples 𝒔 𝟐= standard deviation of Digicel samples
  25. 25. Page | 21 Calculating the Test Statistic - Pooled Estimate of Population Standard Deviation πœŽΜ‚ = √ ( 𝑛1 βˆ’ 1) 𝑠1 2 + ( 𝑛2 βˆ’ 1) 𝑠2 2 𝑛1 + 𝑛2 βˆ’ 2 πœŽΜ‚ = √ (15 βˆ’ 1)13.522 + (14 βˆ’ 1)11.142 15 + 15 βˆ’ 2 πœŽΜ‚ = √ 4296.46 28 πœŽΜ‚ = 12.38729188 - Standard error for the sampling distribution of 𝑿̅ 𝟏 βˆ’ 𝑿̅ 𝟐 𝜎 𝑋1 βˆ’π‘‹2 = 𝜎.Μ‚ √ 1 𝑛1 + 1 𝑛2 𝜎 𝑋1βˆ’π‘‹2 = 12.38729188√ 1 15 + 1 15 𝜎 𝑋1βˆ’π‘‹2 = 4.523199459 - Test Statistic 𝑑 = ( 𝑋̅1 βˆ’ 𝑋̅2) βˆ’ ( πœ‡1 βˆ’ πœ‡2) 𝜎 𝑋1βˆ’π‘‹2 𝑑 = (βˆ’76.607 βˆ’ βˆ’67.647) βˆ’ 0 4.523199459
  26. 26. Page | 22 NB: πœ‡1 βˆ’ πœ‡2 is 0 because it is assumes that the population means are the same. 𝑑 = βˆ’8.96 4.523199459 𝑑 = βˆ’1.981 Conclusion Remember that we would only reject the null hypothesis if t was outside the acceptance region -2.052 < t < 2.052. However, since t fails to meet the criteria for rejection, we do not have sufficient evidence to reject the null hypothesis. Hence we fail to reject then null hypothesis and conclude that LIME and Digicel coverage has no difference in Vide-Bouteille.
  27. 27. Page | 23 Mathematical Knowledge Measures of Central Tendency The mean of ten readings was used as an average measure of signal strength in each of the 15 areas. This specific measure of central tendency was used because it takes every single reading into account. The mean was a suitable measure for the average reading for each area because the readings for each area fluctuated within a reasonably small range. The mean coverage in Vide-Bouteille for each company was also found by taking the mean of all the area means. However, this measure is not the best representation of the overall signal strength for each company in Vide-Bouteille due to the presence of outliers amongst the data. For some areas there were exceptionally good readings whilst for other areas there were critically bad readings. Hence it was more logical to take the mean to represent the overall signal strength for each company. The median readings of the two companies – LIME (-75.8) and Digicel (-70.7) – were much closer than the two means – LIME (-76.6) and Digicel (-67.6). This is because the LIME mean was affected by a few negative outliers whilst the Digicel was affected by some positive outliers. Measures of Spread The standard deviation of the LIME and Digicel readings was used to measure how consistent the LIME and Digicel readings were. Standard deviation measures the average distance of a set of data from the mean. Both sets of data were fairly consistent, with the Digicel readings being slightly more clustered around it mean. The inter-quartile ranges were also used to measure the spread of the data. Whilst the standard deviation measures the spread of the entire data, the inter-quartile range represents the middle 50% of the data. Hypothesis Test Though one may point out Digicel as the better coverage provider in Vide-Bouteille, it must be noted that the mean signal strengths obtained were only samples. Only 15 areas in Vide- Bouteille were tested; there were many more areas which could have been tested. If these areas were tested, it could have possibly made a difference in the mean signal strengths. Hence one cannot make a concrete conclusion on a sample. This is where a Hypothesis test comes in.
  28. 28. Page | 24 Using the empirical rule, we know that about 95% of samples collected from the population will have a sample mean that falls within 2 standard deviations from the population mean. It is therefore unlikely (less than a 5% probability) that we select a sample beyond 2 standard deviations from the mean, if the population mean is indeed correct. Null and Alternative Hypothesis It was assumed that the mean signal strengths of LIME and Digicel in Vide-Bouteille were the same, i.e H0: 𝑿1 – 𝑿2 = 0 This was called the null hypothesis. Hence, assuming that this was true, the majority of sample mean differences should be within the acceptance region (set at a 95% confidence level.) Thus, if a sample actually fell outside the acceptance region, we should reject the null hypothesis and conclude that the mean signal strengths are not the same; ie. Ha: 𝑿1 – 𝑿2 β‰  0 This was the alternative hypothesis. If the sample means fell beyond a certain distance from the population means difference, we will reject the null hypothesis in favor on this hypothesis. Confidence / Significance Level The hypothesis test was set at a confidence level of 95% - we were 95% sure that that LIME and Digicel coverage were the same in Vide-Bouteille. Thus we were 5% sure that this was not true. 5% (0.05) is called the level of significance or the alpha level. The empirical rule applies for only a normally distributed sample but for the hypothesis test, a t- distribution was used for the data which is very similar to a normal distribution. For the t- distribution for this population 95% of the sample means differences fell between 2.052 SD’s of the population means difference. These values are called the critical values. The range >2.052 t < -2.052 is called the critical region or rejection region. If any mean difference fell beyond 2.052 SD’s, then would assume that the LIME and Digicel coverage are not the same in Vide-Bouteille. T- Score
  29. 29. Page | 25 After the criterion was set we needed a value to test whether the data fit the criteria or didn’t fit the criteria. This is called the t-value or t-score or test statistic. It is essentially, the number of standard deviations of the sample away from the mean.
  30. 30. Page | 26 Analysis Both companies had areas which were excellently serviced, some poorly serviced, whilst other received average reception. Several factors can be held responsible for the different readings, but no single factor can be said to have a dominating influence. Some areas received better reception from a certain site because their cell sites were closer but some areas received better reception from a company’s cell site which was further away than its competitor’s own – probably because of the cell site frequency, or because dense vegetation or land structures, or because the elevation of the cell site antenna. This shows that so many factors can have a big influence on cell phone reception. One area in particular had extremely bad reception; sometimes receiving no reception at all. This was for LIME coverage in Area 13. One may blame the fact that it is 1.5km away from the tower, but LIME phones have received better coverage from towers further than this. Looking closer however, google earth was used to show an elevation profile from the area to the cell site and it was found that a densely vegetated 82m hill spanning 300m across the ground was obstructing the line of sight from the cell site to the area. There was a LIME tower only 700m away from Area #13 so why didn’t the area receive its coverage from this cell site? This is because a 40m was directly blocking the site’s line of sight from the area. Furthermore, the cell tower was not elevated in terms of the area; it was approximately 40 m lower. There were two areas which received exceptionally good reception however. Let us first investigate Area #1 which received a mean signal strength of -47.7dB from a Digicel tower. It was extremely close to the cell site, - just 263m away, and there were no large obstructions. Furthermore, the cell tower was a powerful one, with a frequency of 900MHz. Area #5 received a mean signal strength of -48.1 dB from LIME and just like Area#1 for Digicel, it was extremely close to the tower; just 354m away. Likewise, there were no huge obstructions to the line of sight and the power was a powerful one – a dual band of 850MHz and 1900 MHz. Most likely the phone was receiving signal from the more powerful 850MHz channel. The standard deviations of the data for both companies showed that the readings were fairly consistent around Vide-Bouteille. However looking at the spread of the middle 50 % of their distributions, different facts were discovered. There were no positive or negative skews for the LIME data, the upper semi-interquartile range was extremely close to the lower semi- interquartile range. However, for the Digicel data a positive skew was seen. The upper-semi interquartile range was significantly larger than the lower semi-interquartile range. This indicates that the Digicel readings were consistently high.
  31. 31. Page | 27 Both the mean and median measures for the Digicel readings were higher than that of LIME. Furthermore, it was shown that the middle 50% of Digicel readings were consistently high whilst that of LIME showed no indication of a positive skew. All this data would lead one to think that Digicel has better coverage than LIME. However the data was just a sample, and a hypothesis test was used to come to a conclusion. At a confidence level of 95% the conclusion was drawn that LIME and Digicel coverage were the same in Vide-Bouteille. Remember however there is a 5% likelihood that the conclusion is wrong. The preliminary analysis using the measures of central tendency and measures of spread surely challenges the conclusion and indicates that it might actually be wrong. The test statistic only barely fell within the acceptance region. If the hypothesis test was one tailed, and the alternative hypothesis stated that Digicel coverage is better than LIME coverage in Vide-Bouteille, the test statistic would have actually fell into the rejection region. Hence more research should be done in the Vide-Bouteille area to come to a more solid conclusion.
  32. 32. Page | 28 Findings 1. The mean signal strength for LIME phones in Vide-Bouteille is -76.6 dB whilst the median signal strength is -75.8dB. 2. LIME coverage in Vide-Bouteille is fairly consistent with a standard deviation of 14dB. 3. The mean signal strength for Digicel phones in Vide-Bouteille is -67.6 dB whilst the median signal strength is -10.7dB. 4. Digicel coverage in Vide-Bouteille is fairly consistent, with a standard deviation of 11.48dB. 5. Based on the sample of data collected, it was concluded that LIME and Digicel coverage are the same in Vide-Bouteille.
  33. 33. Page | 29 Recommendations 1. LIME should install a micro cell site near Area #5 to improve the critically bad reception that it receives. 2. More research should be done to come to a solid conclusion as to which company provides better cell coverage in Vide-Bouteille.
  34. 34. Page | 30 Conclusion Digicel has both a higher mean cell reception (-67.6dBm) and higher median cell reception (-70.5dBm) in Vide-Bouteille compared to LIME’s mean (-76.6dBm) and median (-75.8dBm). The Digicel data had a positive skew which was clearly illustrated in the box & whiskers plot. This indicates that Digicel readings in Vide-Bouteille are consistently high. However, the Box & Whiskers plot for LIME readings in Vide-Bouteille showed no skew, which indicates that LIME readings in Vide-Bouteille are neither consistently low nor consistently high. Whilst there were 5 Digicel areas which received a mean signal strength greater than -60.5dBm, (excellent reception), only one area yielded such a value for LIME reception. Furthermore, Digicel only had one area which received less than -90.5 dBm (bad reception) whilst LIME had three such areas. Of these 3 areas serviced by LIME, one proved to be a major dead zone. This was Area #13. It received critically bad reception for the first 3 readings then for the next 7 readings no reception was received. Digicel had no dead zones however, all of its areas were relatively well serviced. Simple analysis of all of the above facts may lead one to draw the conclusion that Digicel and LIME coverage are not the same, and actually, that Digicel coverage is better in Vide-Bouteille. However, we must keep in mind that the areas tested were merely a sample. So did that sample provide enough evidence to reject the claim that Digicel and LIMEcoverage are the same, and to accept the claimthat they are different? No it did not. The test statistic fell within the region of acceptance for the null hypothesis which meant that there was not sufficient evidence to reject the null hypothesis that LIME and Digicel coverage are the same in Vide- Bouteille. However, the test statistic almost fell out of the region of acceptance. Keep in mind that there is still a possibility of error in the hypothesis. Hence similar research should be conducted in Vide-Bouteille on a larger scale to get a more powerful conclusion on the question β€œIs LIME and Digicel coverage the same in Vide-Bouteille?” But for now, based on the data presented before us, we can conclude that LIME and Digicel coverage have no difference in Vide-Bouteille.
  35. 35. Appendix & Bibliography
  36. 36. Page | 31 Area Distances from Cell Sites Above: Master Table Showing Mean Readings for Each Area and Their Distance From Each Company’s Cell Sites Area LIME Digicel Mean Tower Frequency (MHz) Ground Distance (m) Tower Elevation (m) Line of Sight (m) Tower Frequency (MHz) Mean Ground Distance (m) Tower Elevation (m) Line of Sight (m) #1 -65.2 850 & 1900 689 70 693 900 -47.7 261 31 263 #2 -64.3 850 & 1900 539 98 548 900 -57.9 287 26 288 #3 -64.9 1900 168 5 168 900 -61.4 555 24 556 #4 -71.9 850 & 1900 2204 11 2204 900 -65.8 2407 8 2407 #5 -48.1 850 & 1900 354 -13 354 900 -77.9 284 62 291 #6 -80.7 1900 158 -24 160 900 -90.9 745 9 745 #7 -75.2 850 & 1900 1054 -34 1053 900 -59.6 2807 223 2816 #8 -84.9 850 & 1900 1845 31 1845 900 -72 3025 257 3036 #9 -84.9 850 & 1900 1470 53 1471 900 -52.7 2688 237 2698 #10 -67.5 850 & 1900 459 -22 460 900 -74.6 1839 19 1839 #11 -86.4 850 & 1900 225 -23 226 900 -81.1 2163 164 2169 #12 -81.3 850 & 1900 2275 7 2275 900 -70.5 2282 7 2282 #13 -107 850 & 1900 1559 60 1560 900 -71.5 2413 158 2418 #14 -91 850 & 1900 2270 23 2270 900 -58.1 2513 170 2519 #15 -75.8 850 & 1900 1197 90 1200 900 -73 872 7 872
  37. 37. Page | 32 How the Line of Sight was Calculated The diagram below shows a right-angled triangle formed by the ground distance, tower elevation and line of sight for Area #1; Where a = ground distance b = tower elevation c = line of sight LIME Tower c b =85m Area #1 a = 555m So using basic Pythagoras’ theorem: π‘Ž2 + 𝑏2 = 𝑐2 𝑐 = √ π‘Ž2 + 𝑏2 𝑐 = √5552 + 852 𝑐 = √315,250 𝑐 = 561π‘š All other line of sight calculations were made in a similar manner.
  38. 38. Page | 33 Bibliography Cell Site. (2013, 3 8). RetrievedfromWikipedia:http://en.wikipedia.org/wiki/Cell_site dBm. (2013, 2 27). RetrievedfromWikipedia:http://en.wikipedia.org/wiki/DBm Duplex (Telecommunications). (2013,02 26). RetrievedfromWikipedia: http://en.wikipedia.org/wiki/Duplex_%28telecommunications%29 Farber,L. &. (2011, January 21). FrederickJGravetter,L. B. (2008). Essentials of Statisticsfor the BehaviouralSciences. Belmont:Cengage Learning. HypothesisTestingforTwoSamples. IntroductiontoHypothesisTesting. Steve Dobbs,J.M. (2010). Advanced LevelMathematic: Statistics1. Cambridge:Cambridge University Press. Steve Dobbs, J.M. (2011). Advanced LevelMathematics:Statistics 2. Cambridge:Cambridge University Press.

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