Exercise 41: Check Your Answers Against Those In The Answer

Exercise 41check Your Answers Against Those In The Answer Sectionlis

Exercise 4.1 involves analyzing data related to the selling prices of vehicles sold by Ray Steele Auto Group in Albuquerque, New Mexico. The tasks include sorting the data from lowest to highest and creating a dot plot to visually represent the data distribution. Exercise 4.2 requires constructing a stem-and-leaf chart based on the miles per gallon recorded for a sample of 20 mid-sized cars equipped with a new engine developed by Jansen Motor Company, aimed at reducing gasoline consumption. Exercise 4.3 involves analyzing the selling prices of 19 lakeside lots in Pinnacle Peak, a vacation home community, by calculating key quartiles (first and third quartiles), the median, and drawing a box plot for the data. Exercise 4.4 pertains to a dealership's biweekly commissions, where the goal is to determine the skewness direction of the distribution and verify it by calculating the coefficient of skewness based on given measures of central tendency and dispersion. Exercise 4.5 analyzes the relationship between fuel tank capacity and cruising range in miles for 6 SUVs, involving creating a scatter diagram and characterizing the nature of their relationship. Exercise 4.6 uses a contingency table to compute percentages related to employees working at home or in the office, segmented by gender. The subsequent questions relate to the normal distribution of pregnancy lengths in animals, with calculations to find probabilities of pregnancies lasting less than a certain number of days for individual pregnancies and sample means using the standard normal distribution (z-scores). The task involves applying properties of the normal distribution, including the calculation of probabilities for a single event and means of samples of sizes 29 and 57, with the proper use of the standard normal table or z-score formulas.

Sample Paper For Above instruction

Introduction

The analysis of data through statistical methods provides critical insights into various aspects of business, science, and everyday phenomena. In this paper, we explore several data analysis exercises involving descriptive statistics, probability, and data visualization techniques. Specifically, we examine vehicle prices, fuel efficiency, real estate values, commission distributions, and biological data related to pregnancy durations. Each exercise demonstrates how to apply fundamental statistical concepts, compute relevant measures, and interpret the results to gain a comprehensive understanding of the underlying data.

Exercise Analysis and Data Visualization

The initial exercise involves sorting vehicle prices and creating a dot plot. Sorting data is a fundamental step that organizes the information, making it easier to identify patterns, outliers, and the overall distribution. For the 20 vehicle prices, organized from lowest to highest, a dot plot visually represents the distribution, highlighting clusters or gaps. Such plots are valuable in small datasets because they show the data points plainly, facilitating quick assessments of spread and concentration. The sorted data and dot plot allow for visual and numerical evaluation of the data's symmetry, skewness, and modality.

Similarly, the second exercise tackles the creation of a stem-and-leaf plot for miles per gallon (mpg) from 20 mid-sized cars using a new engine. Stem-and-leaf charts serve as a compact way to display numerical data, preserving the actual data values while revealing the distribution’s shape. The stems group the data by the leading digits, while the leaves indicate the last digit, providing an immediate visual cue about data clustering and spread. For example, if most mpg values cluster around a certain range, the stem-and-leaf plot will reflect this, illustrating the data's central tendency and variability.

The third exercise addresses real estate prices of 19 lakeside lots. Key measures such as quartiles and the median are calculated to summarize the data’s central tendency and dispersion. The first quartile (Q1) indicates the 25th percentile, the median (Q2) the 50th percentile, and the third quartile (Q3) the 75th percentile. These measures are essential for understanding the data distribution, identifying skewness, and detecting outliers. Additionally, drawing a box plot synthesizes these findings visually, showing the spread of the data, potential outliers, and the skewness of the distribution.

The analysis of the dealership’s commissions in exercise 4.4 involves assessing the shape of the distribution using skewness. Given the mean ($1385), median ($1330), and standard deviation ($75), we can infer that the distribution might be positively skewed because the mean exceeds the median. To confirm, the coefficient of skewness is computed, providing a numerical measure of skewness. This value quantifies the asymmetry of the distribution, which helps in choosing appropriate statistical tests and understanding data behavior.

In exercise 5, the relationship between fuel tank capacity and cruising range in miles for six SUVs is examined. Constructing a scatter diagram visually depicts the correlation between these two variables. Typically, a positive correlation is expected: larger fuel tanks allow for longer cruising ranges. The scatter plot, combined with statistical measures such as correlation coefficients, can then describe whether the relationship is strong, weak, or moderate, and whether any outliers exist that might influence the analysis.

The data in exercise 6 involve percentages related to employees working remotely based on gender. Using the contingency table, percentages are computed to analyze workforce distribution. These calculations help in understanding gender-based employment patterns and can further facilitate discussions on diversity and workplace policies.

Finally, the series of questions about pregnancy durations involve applying the properties of the normal distribution. Calculations of probabilities for pregnancies lasting less than a specific number of days involve computing z-scores based on the mean and standard deviation. For sample sizes of 29 and 57, the Central Limit Theorem allows us to approximate the distribution of the sample mean as normal, using the standard error, and then find the associated probabilities. These exercises highlight fundamental concepts in statistical inference, particularly hypothesis testing and the use of z-scores for probability calculations.

Conclusion

The exercises presented demonstrate the breadth of statistical tools available for data analysis, from descriptive statistics and visualizations to probability calculations and inferential statistics. Whether examining vehicle prices, fuel efficiency, real estate values, or biological data, these methods facilitate meaningful interpretations that can inform business decisions, scientific understanding, and policy formulations. The ability to visualize data effectively, summarize distributions, and perform probability estimates underscores the importance of statistical literacy in diverse fields. Mastery of these concepts equips analysts and researchers with the skills needed to interpret real-world data accurately and confidently.

References

• Kreyszig, E. (2011). _Advanced Engineering Mathematics_ (10th ed.). John Wiley & Sons.
• Devore, J. L. (2015). _Probability and Statistics for Engineering and the Sciences_ (9th ed.). Cengage Learning.
• Montgomery, D. C., & Runger, G. C. (2014). _Applied Statistics and Probability for Engineers_ (6th ed.). Wiley.
• Triola, M. F. (2018). _Elementary Statistics_ (13th ed.). Pearson.
• Ross, S. M. (2014). _Introduction to Probability and Statistics_ (11th ed.). Academic Press.
• Freedman, D., Pisani, R., & Purves, R. (2007). _Statistics_ (4th ed.). W. W. Norton & Company.
• Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). _Mathematical Statistics with Applications_ (7th ed.). Cengage Learning.
• Rice, J. A. (2006). _Mathematical Statistics and Data Analysis_. Pacific Grove, CA: Duxbury.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). _Introduction to the Practice of Statistics_ (7th ed.). W. H. Freeman.
• Newbold, P., Carlson, W. L., & Thall, S. (2018). _Statistics for Business and Economics_ (10th ed.). Pearson.