lesson_questions_Ch_one .edited.docx
- 1. SDUIS BUS 702 Text: Statistical Techniques in Business & Economics, by Lind, Marchal, Wathen; McGraw Hill, 17th ed. Homework Questions - Lesson One Question No. 1 Although statistics is defined as numerical information or a collection of quantitative data, what are the practical uses of statistics? Give three examples. Answer 1. Governmental Departments Government departments use statistical techniques for arising on decisions regarding crimes, finances, population, health, education, etc. Statistical departments conduct research’s for seeking specific information on educational and other welfare matters to check the progress and efficiency of spending in different sectors of the economy. Government Departments use these techniques to forecast and predict future spending and collections of funds. Health and Medicine Statistical techniques are being used in the medical industry to assess the reactions of the human body while testing new developments. It accumulates the outcomes and provides the information for future testing's and its practical uses. The techniques available in statistics also assists the researcher in making decision making regarding specific research's to carry on or to be stopped. Corporate Sector Large corporations use statistical techniques to forecast their sales based on available data and evaluate market trends. Statistical techniques also provide a decision for products to be manufactured or not or how much to produce. Statistics uses in each and every sector, it is economics or welfare. Its uses cannot be sum up, however, it is established that no field is complete without statistics. Statistics is a tool that enhances productivity. Question No. 2 What is the difference between qualitative and quantitative variables? Answer 2. By nature, variables can be of two types qualitative and quantitative. Qualitative variables are descriptive variable, these are related to the quality of data. As red balls, in it, red is the quality of the variable. Qualitative variables can influence the parameters of research that can impact the experiment or research outcomes. Qualitative research is to be inductive and specially used in social research. On the other side quantitative variables is related to the quantity of any variable. These are numbers and figures. These numbers and figures can be based on the quality of variables as red balls in the
- 2. population are qualitative variables and the number of balls is the quantity of data. The quantitative variable can be the percentage of something, any measurement or any number of the unit. It is used in economic data especially. Question No. 3 What are the differences between a sample and a population? Answer 3. The population represents the complete set of data based on variables of persons, units, objects and anything that is capable of being considered, based on definite assertions. The population is the complete data which includes a complete amount of data including all elements. However, a sample is the data from the population derived using specific techniques to be assessed and evaluates on specific techniques and reach on the outcome for representing the whole data. Question No. 4 Hershey’s chocolate has been making Hershey’s chocolate bars and Hershey’s kisses for more than 100 years. What data would they collect and how would they use the data to determine if they should continue producing Hershey’s chocolates? Answer 4. Hershey supposed to keep and accumulate the data regarding sales over the past period to simply forecast its future sales. However, if it is possible Hershey supposed to collect the data regarding consumer choice and tastes to forecast numbers close to actual. It also needs to collect the data regarding the buying power of the consumer and to assess the potential number of customers. These specific data are required to reach at precise points from the side of sales. From the production point of view, Hershey's go for inflation rate over the past period and fluctuations in the cost of raw materials. However, government policies couldn't be statistically measured regarding the specific nature of businesses, but their estimations can be created. These are some of the relevant data which need to be collected for deciding between continuing or discontinuing the production process. Question No. 5. How is data measured? Answer 5. Data has four different levels of measurement which are nominal, ordinal, interval and ratio. In the nominal level of measurement, the numbers in the variables are used to classify the data. In this level symbols, letters, signs or words can be used for measurement. For example, gender data can be classified as M for Men F for Female and T for transgender. The second level of measurement is the ordinal level. It creates
- 3. a relationship among data based on some order ascending or descending. For example, grades for students in class reveals the level of standing in the class if ordered by the highest to lowest. Interval level of measurement if the third level of measurement in which intervals are created based on an equivalency, distances among the interval values and measurement. The fourth level of measurement is the ratio level of measurement in which data have equal intervals with properties of having zero value. Among all these levels of measurements, the nominal level is simply used to classify data, whereas the levels of measurement described by the interval level and the ratio level are much more exact. Question No. 6 Lesson Two You are the manager of Joe’s Fast Gas Station. Joe has asked you to follow the learning objectives for chapter two (LO 1 - LO 6) and make a recommendation to increase sales), based on the number of customers who buy gas, and the number of those customers who also buy coffee or a snack (make your own data). Answer 6. Name of Month Number of Customer Visits Number of events when Fuel Sold Number Customer Buys Coffee or Snack January 150 138 85 February 146 140 80 March 161 152 70 April 168 162 68 May 158 155 66 June 149 140 55 July 171 142 52 August 158 148 60 September 144 136 65 October 140 126 72 November 128 116 78 December 118 106 90 Total During the Year 1791 1661 841 The data of Joe's Fast Gas station depicts that customer who buys gas are strictly related to the number of customers who visits its Gas Station. To enhance its sales of gas he has to increase the number of customers by providing maximum capacity and availability of gas filling machines. While looking at the data of customers who buy
- 4. coffee or snack reveals that the purchases are dependent on customer visits but these are also influenced by the season. In the summer season, the sales of coffee are low however in the winter season the coffee sales are high in relation to the number of customer visits to enhance the number of coffee customers Joe should increase the flavors of the coffee. Learning Objectives Learning Objective 1. Summarize qualitative variables with frequency and relative frequency tables. According to Joe’s Gas station data, data can be summarized in grouped form as follow: Name of Month Number of Customer Visits No of Events when Fuel Sold Hi-Octane Diesel Number of Coffee or Snack Sold January 150 138 40 98 85 February 146 140 45 95 80 March 161 152 44 108 70 April 168 162 49 113 68 May 158 155 48 107 66 June 149 140 44 96 55 July 171 142 43 99 52 August 158 148 48 100 60 September 144 136 39 97 65 October 140 126 36 90 72 November 128 116 36 80 78 December 118 106 28 78 90 Total During the Year 1791 1661 500 1161 841 Learning Objective 2. Display a frequency table using a bar or pie chart. Answer:
- 5. Learning Objective 3. Summarize quantitative variables with frequency and relative frequency distributions. Answer: Total Number of Customers who visits Gas Station Fuel Sold Packs of Coffee or Snacks Sold X F Relative Frequency F Relative Frequency 110-120 106 0.06 90 0.10702 120-130 116 0.07 78 0.09275 130-140 126 0.08 72 0.08561 140-150 554 0.33 285 0.33888 150-160 303 0.18 126 0.14982 160-170 314 0.19 138 0.16409 170-180 142 0.09 52 0.06183 1661 1 841 1 Learning Objective 4. Display a frequency distribution using a histogram or frequency polygon Answer 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Number of Customer Visits Events Fuel Sold Hi-Octane Sold Diesel Sold Number Customer Buys Coffee or Snack Sales During the Year
- 6. Lesson Three Using the data you created in Lesson Two, do LO 2,3,4, 5, and 10. Learning Objectives of the Chapter 3 are as follows: LO3-1 Compute and interpret the mean, the median, and the mode. LO3-2 Compute a weighted mean. LO3-3 Compute and interpret the geometric mean. LO3-4 Compute and interpret the range, variance, and standard deviation. LO3-5 Explain and apply Chebyshev’s theorem and the Empirical Rule. Answer Learning Objective 1 X F fx January 118 106 12508 February 128 116 14848 March 140 126 17640 April 144 136 19584 May 146 140 20440 June 149 140 20860 July 150 138 20700 August 158 155 24490 September 158 148 23384
- 7. October 161 152 24472 November 168 162 27216 December 171 142 24282 1661 250424 Mean = x ̄ = ∑ fx/n = 250424/1661 = 150.767 Mean computes the middle value in the data. However the 150.767 is the middle value among the data. X F Cumulative Frequency 110-120 106 106 120-130 116 222 130-140 126 348 140-150 554 902 150-160 303 1205 160-170 314 1519 170-180 142 1661 Median = x ̃ = L+(n/2)-B/G x W = 140+ (1661/2)-348 / 554 x 10 = 148.709 Median Value for the data is 148.709 slightly lower than mean valu which means the dispersion is positively skewed. Mode X F Cumulative Frequency 110-120 106 106 120-130 116 222 130-140 126 348 140-150 554 902 150-160 303 1205 160-170 314 1519 170-180 142 1661
- 8. Mode = l+h(fm – f1 ) / (2 fm – f1 – f2 ) = 140+10(554-126) / (2(554) – 126 – 303) = 146.303 Mode of the data is 146.303 which depicts that the dispersion of the data is positively skewed because it is lower than the mean value. Learning Objective 2 Compute Weighted Mean X January 118 February 128 March 140 April 144 May 146 June 149 July 150 August 158 September 158 October 161 November 168 December 171 X ̄ w = ∑(wx)/∑w = 118+128+140+144+146+149+150+2(158)+161+168+171 / 12 = 1791/12 = 149.25 LO3-3 Compute and interpret the geometric mean. Geometric mean computes the rate of change over the period of time which could be computed from other assumed data
- 9. Year Increase in Air Pollution in Gotham City 1991 2% 1992 1% 1993 3% 1994 4% 1995 5% GM = n√(x1) (x2) (x3) (x4)……… (xn) GM = 10√(1.02)(1.01)(1.03)(1.04)(1.05) GM = 10√1.158728 GM = 1.014841 The GM depicts that about 1.48% increase in rated of pollution in Gotham City averagely over the period of 5 years. LO3-4 Compute and interpret the range, variance, and standard deviation Range computes dispersion in the data so Name of Month X January 150 February 146 March 161 April 168 May 158 June 149 July 171 August 158 September 144 October 140 November 128 December 118 Range = Maximum value – Minimum Value = 171-118 = 53
- 10. Range predicts that the data of visits of customer at Joes Fast Gas Station dispersed at 53. It shows the no. of customers visits can varies in total of 53 values in total. Variance Name of Month X x-µ Squared Deviation January 150 0.75 0.5625 February 146 -3.25 10.5625 March 161 11.75 138.0625 April 168 18.75 351.5625 May 158 8.75 76.5625 June 149 -0.25 0.0625 July 171 21.75 473.0625 August 158 8.75 76.5625 September 144 -5.25 27.5625 October 140 -9.25 85.5625 November 128 -21.25 451.5625 December 118 -31.25 976.5625 1791 2668.25 Variance = ∑(x-µ)/N Variance = 2668.25/12 Variance = 222.345 Standard Deviation Standard Deviation = √∑(x-µ)/n-1 Standard Deviation = √2668.25/11 Standard Deviation = 15.5746 Explain and apply Chebyshev’s theorem and the Empirical Rule. The Russian mathematician P. L. Chebyshev (1821–1894) developed a theorem that allows us to determine the minimum proportion of the values that lie within a specified number of standard deviations of the mean. Chebyshev’s theorem, explains that at least three out of every four, or 75%, of the values must lie between the mean plus two standard deviations and the mean minus two standard deviations. This relationship applies regardless of the shape of the distribution. Further, at least eight of nine values,
- 11. or 88.9%, will lie between plus three standard deviations and minus three standard deviations of the mean. At least 24 of 25 values, or 96%, will lie between plus and minus five standard deviations of the mean. Chebyshev’s Theorem = 1 – 1/K2 At Joes Fast Gas Station has standard deviation of 15.5746 around mean no. of customers visits 149.25. To determine at least what percent of customer visits lie around the 3 standard deviations. To Compute this, we use Chebyshev’s theorem Chebyshev’s Theorem = 1 – 1/K2 Chebyshev’s Theorem = 1 – 1/(3)2 Chebyshev’s Theorem = 1 – 0.11111111 Chebyshev’s Theorem = 88.89% values Empirical Rule Under a frequency distribution, Empirical rule explains the standard deviation through bell shaped graphical representation and scattered of data on the diagram that 68% of the observations will lie within plus and minus one standard deviation of the mean value; about 95% of the observations will lie within plus and minus two standard deviations of the mean; and practically all (99.7%) will lie within plus and minus three standard deviations of the mean.