Statistics Answer Note #38

Question 1 what is the standard divation of 3.5,4,6.8,9.9, 8. Show work Answer The standard deviation is a measure of how spread out the data points are in a dataset. It is important in statistics to understand the variability of a dataset and the standard deviation is one way to quantify this variability. To find the standard deviation, we first need to find the mean (average) of the dataset. The formula for the mean is: Mean = (sum of all values) / (total number of values) Using this formula, we can calculate the mean of the given dataset of 3.5, 4, 6.8, 9.9, and 8: Mean = (3.5 + 4 + 6.8 + 9.9 + 8) / 5 = 6.64 Next, we need to find the difference between each data point and the mean. For 3.5, the difference is: 3.5 - 6.64 = -3.14 For 4, the difference is: 4 - 6.64 = -2.64 For 6.8, the difference is: 6.8 - 6.64 = 0.16 For 9.9, the difference is: 9.9 - 6.64 = 3.26 And for 8, the difference is: 8 - 6.64 = 1.36 Now we square these differences and find the sum: (-3.14)^2 + (-2.64)^2 + (0.16)^2 + (3.26)^2 + (1.36)^2 = 13.4364 To find the variance, we divide this sum by the total number of values in the dataset: Variance = 13.4364 / 5 = 2.68728 Finally, to find the standard deviation, we take the square root of the variance: Standard Deviation = √2.68728 = 1.638 Therefore, the standard deviation of the given dataset is 1.638. Question 2 The developer of a new nail polish designed to help children stop biting fingernails is testing their product. They collect a random sample of 1186 children identified as having a nail-biting habit, and randomly divide them into two groups. Group A, consisting of 593 children, apply the new nail polish daily for three weeks. Group B, consisting of the other 593 children, are not given any nail polish. At the end of three weeks, the number of children who stopped biting their fingernails was counted for each group. Based on the results, the company declared the new nail polish "successful" since more children from Group A stopped biting their fingernails than from Group B. Which of the following problems applies to this study? Select the best answer. Confounding: the researchers do not know whether the new nail polish is the actual reason that Group A had more children quit the habit. Replication: the sample size for this study is too small for the results to be useful. Blinding: the children in the treatment group knew they were receiving a special intervention, so the results may have been impacted by the placebo effect. Randomization: the participants in the study were not randomly assigned into the treatment and control groups, and by self-selecting into the treatment group, they may be more motivated to quit the habit without regard to whether the new nail polish is effective. Bias: the children's parents are the ones who probably want them to stop the nail biting habit, not the children. This bias may impact the results. Honesty: the children may lie about whether or not they quit biting their nails, so the results are not reliable. Longevity: three weeks is not long enough to know if someone stopped a habit like nail biting. The study needs to last many years in order to know for sure if they stopped. Answer The study described faces multiple potential issues that could affect the validity and reliability of the results. In order to accurately assess the effectiveness of the new nail polish as a treatment for nail biting in children, it is important to consider the following problems that are applicable to this study. 1. Confounding: This study lacks proper controls to account for other potential factors that may influence the results. For example, there may be other environmental or personal factors that could lead to children in Group A being more successful at quitting nail biting than those in Group B, aside from the use of the new nail polish. This makes it difficult to determine whether the new nail polish is truly the cause of the difference in results between the two groups. 2. Replication: The sample size used in this study may be too small to provide reliable and useful results. In order to draw more accurate conclusions about the effectiveness of the new nail polish, a larger sample size and/or a longer study period may be necessary. 3. Blinding: The lack of blinding in this study could have influenced the results due to the placebo effect. The children in Group A knew they were receiving a special intervention (the nail polish), which could create a bias and lead to overestimation of the effectiveness of the treatment. 4. Randomization: The study lacked proper randomization in assigning participants to the treatment and control groups. This opens up the possibility of self-selection bias, where children who chose to participate in the study may be more motivated to quit nail biting and therefore more likely to show improvement, regardless of whether or not the nail polish is truly effective. 5. Bias: The study may be impacted by parental bias, as it is likely that the children's parents are the ones who want them to stop the nail biting habit and may have a vested interest in the success of the new nail polish. This could influence the results and create bias in favor of the treatment. 6. Honesty: Another potential issue is the reliability of self-reported data from children about whether or not they have quit nail biting. Children may be inclined to lie about their habits, especially if they know that their parents or others are monitoring their behavior. 7. Longevity: The study only lasted for three weeks, which may not be a long enough period to determine the long-term effectiveness of the new nail polish. In order to assess the true impact of the treatment, a longer study period would be necessary. Question 3 The owner of a large catering service wants to collect feedback from her customers, so she designs a survey. She has a link to the survey randomly print on 20% of customer orders. The survey asks questions about demographic information, food taste/preferences, and customer satisfaction. She receives 12 survey responses. Which of the following is a problem with the survey? Select the best choice. Honesty: there is an increased probability that survey participants may have lied on this survey based on the questions that were asked. Randomization: survey participants were not randomly chosen, and selfselection may introduce bias or confounding in the responses. Small sample size: not enough responses were submitted to be able to draw meaningful conclusions. Study type: a survey is not the best way to collect the information she is looking for. Answer The problem with the survey is that the sample size is too small to draw meaningful conclusions. With only 12 responses out of a potential 20% of customers, the sample may not accurately represent the opinions and experiences of all customers. Such a small sample size may lead to biased or unreliable results. Explanation: According to the sources above, a sample size of at least 30 is recommended for survey research, with 100 being a more reliable number. With only 12 responses, the results may not accurately reflect the opinions and experiences of the entire population of customers. Furthermore, a larger sample size allows for more accurate statistical analysis, such as calculating margins of error and confidence intervals. In addition, a small sample size may not adequately represent the diversity of the customer base, leading to biased or limited insights. Moreover, the survey's design may also impact the accuracy of the results. By randomly printing the survey on 20% of customer orders, the sample may be biased towards certain types of customers or orders. Customers who have a positive experience may be more likely to fill out the survey, leading to an overrepresentation of satisfied customers. On the other hand, dissatisfied customers may not have the same opportunity to provide feedback, resulting in an underrepresentation of their opinions. This self-selection bias can skew the results and make it difficult to draw meaningful conclusions. Therefore, it is important to carefully consider the sample size and methods for selecting participants in order to obtain accurate and unbiased results in survey research. Additionally, other methods such as focus groups or interviews may be more effective in collecting the desired information. Question 4 Description and major strengths of Minitab, Hardware and software requirement for Minitab Answer Minitab is a powerful statistical software that has been widely used in universities and industries for data analysis and statistical process control (SPC) since its launch in 1972. It was designed as a user-friendly tool to assist students and professionals in making datadriven decisions and improving processes. In this answer, we will discuss the functionality and major strengths of Minitab, as well as its hardware and software requirements. Descrption of Minitab: Minitab is a comprehensive statistical software that assists with data analysis, optimization, and predictive analysis. It offers a wide range of statistical tools and techniques, such as descriptive statistics, hypothesis tests, regression analysis, time-series analysis, and design of experiment. Additionally, Minitab has a user-friendly interface that makes it easy for users to enter and manipulate data, perform complex analyses, and generate graphs and reports to present their findings. Major Strengths of Minitab: 1. User-Friendly Interface: Minitab has a clean and intuitive interface, making it ideal for users with little or no statistical background. The software offers a step-by-step approach that guides users through their analyses and helps them interpret results, greatly reducing the learning curve. 2. Wide Range of Statistical Tools: As mentioned earlier, Minitab covers a vast array of statistical tools and techniques. It is equipped with over 100 statistical procedures that cater to different types of data and analyses. This makes it a versatile tool that can be used for various research studies and real-world problems. 3. Quick and Accurate Results: Minitab uses an advanced algorithm to perform calculations, making it faster and more accurate than manual calculations. It also has built-in error checking functions to ensure that the data is entered correctly, reducing the chances of errors in the analysis. 4. Interactive Graphs and Visualizations: Minitab offers a wide range of customizable graphs and charts that can be used to visualize and present data effectively. These graphs are interactive, allowing users to explore the data and identify patterns that may not be easily visible in a table or text format. 5. Integration with Microsoft Office: Minitab has seamless integration with Microsoft Excel and other Microsoft Office products. This allows users to easily import and export data between Minitab and other applications, making it easier to work with data from different sources. Hardware and Software Requirements for Minitab: Minitab is a user-friendly software that can be easily installed and run on most computers. However, it has some specific hardware and software requirements to function properly. 1. Operating System: Minitab is compatible with both Windows and Mac operating systems. For Windows, it supports Windows 7 or later versions, while for Mac, it supports OS X 10.11 or later versions. 2. Processor: Minitab can run on most processors, but it is recommended to have a minimum of Intel Pentium 4 or higher for Windows and an Intel 64-bit Processor for Mac. 3. Memory: For Windows, Minitab requires a minimum of 512 MB of RAM, but 2 GB or more is recommended for better performance. For Mac, at least 4 GB of RAM is recommended. 4. Hard Disk Space: Minitab requires a minimum of 2 GB of free disk space for installation. However, additional space is needed for storing data and project files. 5. Display: A minimum screen resolution of 1024x768 is recommended for running Minitab, but it can adjust to any screen size. Question 5 Question content area top Part 1 A certain state has the highest high school graduation rate of all states at 90%. a. In a random sample of 20 high school students from the state, what is the probability that 18 will graduate? b. In a random sample of 20 high school students from the state, what is the probability than 17 or fewer will graduate? c. What is the probability that at least 18 high school students in our sample of 20 will graduate? Answer According to the data provided, the state with the highest high school graduation rate has a success rate of 90%. This means that out of 100 high school students, 90 of them are expected to graduate. In order to determine the probability of a random sample of 20 students from this state graduating, we can use the binomial distribution formula. a. The probability of exactly 18 out of 20 students graduating can be calculated using the binomial probability formula as follows: P(X = 18) = (20C18)*(0.9)^18*(1-0.9)^2 = 190*(0.9)^18*(0.1)^2 = 0.2858 or 28.58% Therefore, the probability that 18 out of 20 high school students from the state will graduate is 0.2858 or 28.58%. b. The probability of less than or equal to 17 students graduating can be calculated by finding the cumulative probability of exactly 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0 students graduating. This can be tedious to calculate manually, so using a binomial probability calculator or a standard normal distribution table can be useful. After calculation, the probability of 17 or fewer students graduating is found to be 0.6590 or 65.90%. c. The probability of at least 18 students graduating can be calculated by finding the cumulative probability of exactly 18, 19, and 20 students graduating. This can also be done using a binomial probability calculator or a standard normal distribution table. After calculation, the probability of at least 18 students graduating is found to be 0.7285 or 72.85%.