Comparing two groups #1 - Questions and Answers

Question 1 Why is it important that a sample be random and representative when conducting hypothesis testing? Representative Sample vs. Random Sample: An Overview Economists and researchers seek to reduce sampling bias to near negligible levels when employing statistical analysis. Three basic characteristics in a sample reduce the chances of sampling bias and allow economists to make more confident inferences about a general population from the results obtained from the sample analysis or study. Such samples must be representative of the chosen population studied. They must be selected at random, meaning each member of the larger population has an equal chance of being chosen. They must be big enough so as not to bias the results. The optimal size of the sample group depends on the precise degree of confidence required for making an inference. Representative sampling and random sampling are two techniques used to help ensure data is free of bias. These sampling techniques are not mutually exclusive and, in fact, they are often used in tandem to reduce the degree of sampling error in an analysis and allow for greater confidence in making statistical inferences from the sample in regard to the larger group. Representative Sample: A representative sample is a group or set chosen from a larger statistical population or group of factors or instances that adequately replicates the larger group according to whatever characteristic or quality is under study. A typical sample mirrors important variables and traits of the large society being studied. Some examples include sex, age, education level, socioeconomic status (SES), or marital status. A larger sample size reduces sampling error and increases the likelihood that the sample accurately reflects the target population. Random Sample: A random sample is a group or set chosen from a larger population or group of factors or instances in a random manner that allows for each member of the larger group to have an equal chance of being chosen. A random sample is meant to be an unbiased representation of the larger population. It is considered a fair way to select a sample from a larger population since every member of the population has an equal chance of getting selected. Special Considerations: People collecting samples need to ensure that bias is minimized. Representative sampling is one of the key methods of achieving this because such samples replicate as closely as possible elements of the larger population under study. This alone, however, is not enough to make the sampling bias negligible. Combining the random sampling technique with the representative sampling method reduces bias further because no specific member of the representative population has a greater chance of selection into the sample than any other. Summarize this article in 250 words. Answer Summary of the Article: This article states about Sample and what things are important to consider when we conduct a hypothesis testing for our research. A sample is very important in research because it is a set of individuals, units, or objects which researcher collect from a population. Since, Researcher can not collect data and test their hypothesis from whole population so they take sample by defined procedure to interence their results and then test their hypothesis. The two techniques that help researcher to get reliable and error free results are random and representative sampling techniques. Random sampling is a technique in which sample is collected from population randomly. In this technique, the sample is more true, free from bias. This is because random sampling make sure each person or object have fair and equal chance to be chosen. Representative technique, as the name suggests, is a technique that accurately shows or represents the characteristics of a large population. The characteristics like age, gender, marital status, education level etc. for example - If we study the behavior of poor results of students in Maths and science then we collect sample. The class have 40 students where 20 are male and 20 are female students so we collect sample of 10 students where 5 represent male and 5 represent female students. The purpose of every study is to diminish the probability of containing biases and mistakes. Using both techniques will help researcher to achieve results with less bias and sampling error so that hypothesis testing can give real and true result. Question 2 What is an advantage of using a stem-and-leaf plot instead of a histogram? What is a disadvantage? Answer 1 Advantage: Stem-and-leaf plots show original data values where histograms do not. One downside of histograms is that they can easily organize data of all sizes, whereas stem-and-leaf plots cannot. Answer 2 The advantage of using a stem-and-leaf plot instead of a histogram is that it provides a more detailed representation of the data, particularly for smaller data sets. In a stem-and-leaf plot, each data point is displayed individually, allowing for a direct visualization of the actual values. This level of granularity can be useful when analyzing the distribution and identifying specific data points. On the other hand, a disadvantage of using a stem-and-leaf plot is that it can become cumbersome and difficult to interpret when dealing with larger data sets. As the number of data points increases, the stem-and-leaf plot can become overcrowded and cluttered, making it challenging to extract meaningful insights. In such cases, a histogram may be more suitable as it summarizes the data into distinct intervals, providing a clearer overview of the distribution. Question 3 In a one-way ANOVA, what does it mean to reject the statement in the null hypothesis if three treatment groups are being compared? Answer The null hypothesis for one-way ANOVA: 𝐻 0 : 𝜇 1 = 𝜇 2 = 𝜇 3 H 0 ​ :μ 1 ​ =μ 2 ​ =μ 3 ​ 𝐻 1 : 𝜇 1 ≠ 𝜇 2 ≠ 𝜇 3 H 1 ​ :μ 1 ​  =μ 2 ​  =μ 3 ​ Where 𝜇 1 , 𝜇 2 , μ 1 ​ ,μ 2 ​ , and 𝜇 3 μ 3 ​ are means of three different groups respectively. If the null hypothesis is rejected, it means that one of the means is different from another two means. Hence, rejecting the statement in the null hypothesis means that there is sufficient evidence to conclude that the population mean is different than the other two means or that all three population means are different. Question 4 To describe: The reason for the name "gravity model." Answer Given: The number of daily phone calls between two cities varies jointly as their populations and varies inversely as the square of the distance between them. The above stated model is called as gravity model. Definition used: (1) Joint Variation: Joint variation is a variation in which a variable varies directly as the product of two or more other variables. Thus, the equation 𝑦 = 𝑘 𝑥 𝑧 y=kxz, is read y varies jointly as x and z. (2) Inverse Variation: If a situation is described by an equation in the form 𝑦 = 𝑘 𝑥 y= x k ​ , where k is a nonzero variation constant, then it can be said that y varies inversely as x or y is inversely proportional to x. Description: Let the average number of daily phone calls is represented as 𝐶 C and population of two cities are 𝑃 1 P 1 ​ and 𝑃 2 P 2 ​ , respectively. The distance between two cities be represented as 𝑑 d. From definition (1), the given "The C between the two cities varies jointly as the product of their population" can be expressed as, 𝐶 = 𝑘 𝑃 1 𝑃 2 . C=kP 1 ​ P 2 ​ . Similarly by the definition (2) the given "The C varies inversely as the as the square of distance" can be expressed as, 𝐶 = 𝑘 𝑑 2 . C= d 2 k ​ . Combine the equation (1) and equation (2) as follows. 𝐶 = 𝑘 𝑃 1 𝑃 2 𝑑 2 . C= d 2 kP 1 ​ P 2 ​ ​ . From equation (3), the given model can be expressed as 𝐶 = 𝑘 𝑃 1 𝑃 2 𝑑 2 C= d 2 kP 1 ​ P 2 ​ ​ , where k is the constant of variation. In general, the newton's formula for gravitation is 𝐹 = 𝐺 𝑚 1 𝑚 2 𝑑 2 . F= d 2 Gm 1 ​ m 2 ​ ​ . The above formula indicates the gravitational force between two objects in the universe. On comparing the gravitational formula with given model, it is observed to be in similar form of joint variation. Thus, the given model is called as gravity model.