### 统计代写|商业分析作业代写Statistical Modelling for Business代考|Random Sampling, Three Case Studies

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|商业分析作业代写Statistical Modelling for Business代考|Random sampling

If the information contained in a sample is to accurately reflect the population under study, the sample should be randomly selected from the population. To intuitively illustrate random sampling, suppose that a small company employs 15 people and wishes to randomly select two of them to attend a convention. To make the random selections, we number the employees from 1 to 15 , and we place in a hat 15 identical slips of paper numbered from 1 to 15 . We thoroughly mix the slips of paper in the hat and, blindfolded, choose one. The number on the chosen slip of paper identifies the first randomly selected employee. Then, still blindfolded, we choose another slip of paper from the hat. The number on the second slip identifies the second randomly selected employee.

Of course, when the population is large, it is not practical to randomly select slips of paper from a hat. For instance, experience has shown that thoroughly mixing slips of paper (or the like) can be difficult. Further, dealing with many identical slips of paper would be cumbersome and time-consuming. For these reasons, statisticians have developed more efficient and accurate methods for selecting a random sample. To discuss these methods we let $n$ denote the number of elements in a sample. We call $n$ the sample size. We now define a random sample of $n$ elements and explain how to select such a sample ${ }^{2}$.

In making random selections from a population, we can sample with or without replacement. If we sample with replacement, we place the element chosen on any particular selection back into the population. Thus, we give this element a chance to be chosen on any succeeding selection. If we sample without replacement, we do not place the element chosen on a particular selection back into the population. Thus, we do not give this element a chance to be chosen on any succeeding selection. It is best to sample without replacement. Intuitively, this is because choosing the sample without replacement guarantees that all of the elements in the sample will be different, and thus we will have the fullest possible look at the population.

We now introduce three case studies that illustrate (1) the need for a random (or approximately random) sample, (2) how to select the needed sample, and (3) the use of the sample in making statistical inferences.

## 统计代写|商业分析作业代写Statistical Modelling for Business代考|Selecting a Random Sample

Part 2: Selecting a Random Sample The first step in selecting a random sample is to obtain a numbered list of the population elements. This list is called a frame. Then we can use a random number table or computer-generated random numbers to make random selections from the numbered list. Therefore, in order to select a random sample of 100 employees from the population of 2,136 employees on 500 -minute-per-month cell phone plans, the bank will make a numbered list of the 2,136 employees on 500 -minute plans. The bank can then use a random number table, such as Table 1.4(a) on the next page, to select the random sample. To see how this is done, note that any single-digit number in the table has been chosen in such a way that any of the single-digit numbers between 0 and 9 had the same chance of being chosen. For this reason, we say that any single-digit number in the table is a random number between 0 and 9 . Similarly, any two-digit number in the table is a random number between 00 and 99 , any three-digit number in the table is a random number between 000 and 999 , and so forth. Note that the table entries are segmented into groups of five to make the table easier to read. Because the total number of employees on 500 -minute cell phone plans $(2,136)$ is a four-digit number, we arbitrarily select any set of four digits in the table (we have circled these digits). This number, which is 0511 , identifies the first randomly selected employee. Then, moving in any direction from the 0511 (up, down, right, or left-it does not matter which), we select additional sets of four digits. These succeeding sets of digits identify additional randnmly selected emplnyees. Here we arbitrarily move down from 0511 in the table. The first seven sets of four digits we obtain are
$\begin{array}{lllllll}0511 & 7156 & 0285 & 4461 & 3990 & 4919 & 1915\end{array}$
(See Table 1.4(a) – these numbers are enclosed in a rectangle.) Because there are no employees numbered $7156,4461,3990$, or 4919 (remember only 2,136 employees are on 500 -minute plans), we ignore these numbers. This implies that the first three randomly selected employees are those numbered 0511, 0285, and 1915. Continuing this procedure, we can obtain the entire random sample of 100 employees. Notice that, because we are sampling without replacement, we should ignore any set of four digits previously selected from the random number table.

While using a random number table is one way to select a random sample, this approach has a disadvantage that is illustrated by the current situation. Specifically, because most fourdigit random numbers are not between 0001 and 2136 , obtaining 100 different, four-digit random numbers between 0001 and 2136 will require ignoring a large number of random numbers in the random number table, and we will in fact need to use a random number table that is larger than Table 1.4(a). Although larger random number tables are readily available in books of mathematical and statistical tables, a good altemative is to use a computer

software package, which can generate random numbers that are between whatever values we specify. For example, Table $1.4$ (b) gives the Minitab output of 100 different, four-digit random numbers that are between 0001 and 2136 (note that the “leading 0 ‘ $\mathrm{s}$ ” are not included in these four-digit numbers). If used, the random numbers in Table 1.4(b) would identify the 100 employees that form the random sample. For example, the first three randomly selected employees would be employees 705,1990 , and $1007 .$

Finally, note that computer sofware packages sometimes generate the same random number twice and thus are sampling with replacement. Because we wished to randomly select 100 employees without replacement, we had Minitab generate more than 100 (actually, 110) random numbers. We then ignored the repeated random numbers to obtain the 100 different random numbers in Table $1.4$ (b).

## 统计代写|商业分析作业代写Statistical Modelling for Business代考|Rating a Bottle Design

Part 1: Rating a Bottle Design The design of a package or bottle can have an important effect on a company’s bottom line. In this case a brand group wishes to research consumer reaction to a new bottle design for a popular soft drink. Because it is impossible to show the new bottle design to “all consumers,” the brand group will use the mall intercept method to select a sample of 60 consumers. On a particular Saturday, the brand group will choose a shopping mall and a sampling time so that shoppers at the mall during the sampling time are a representative cross-section of all consumers. Then, shoppers will be intercepted as they walk past a designated location, will be shown the new bottle, and will be asked to rate the bottle image. For each consumer interviewed, a bottle image composite score will be found by adding the consumer’s numerical responses to the five questions shown in Figure 1.4. It follows that the minimum possible bottle image composite score is 5 (resulting from a response of 1 on all five questions) and the maximum possible bottle image composite score is 35 (resulting from a response of 7 on all five questions). Furthermore, experience has shown that the smallest acceptable bottle image composite score for a successful bottle design is 25 .

Part 2: Selecting an Approximately Random Sample Because it is not possible to list and number all of the shoppers who will be at the mall on this Saturday, we cannot select a random sample of these shoppers. However, we can select an approximately random sample of these shoppers. To see one way to do this, note that there are 6 ten-minute intervals during each hour. and thus there are 60 ten-minute intervals during the 10-hour period from 10 A.M. to 8 P.M. – the time when the shopping mall is open. Therefore, one way to select an approximately random sample is to choose a particular location at the mall that most shoppers will walk by and then randomly select – at the beginning of each ten-minute period-one of the first shoppers who walks by the location. Here, although we could randomly select one person from any reasonable number of shoppers who walk by, we will (arbitrarily) randomly select one of the first five shoppers who walk by. For example, starting in the upper left-hand corner of Table 1.4(a) and proceeding down the first column, note that the first three random numbers between 1 and 5 are 3,5 , and 1 . This implies that ( 1 ) at 10 A.M. we would select the 3 rd customer who walks by; (2) at $10: 10$ A.M. we would select the 5 th shopper who walks by; (3) at 10:20 A.M. we would select the 1 st customer who walks by, and so forth. Furthermore, assume that the composite score ratings of the new bottle design that would be given by all shoppers at the mall on the Saturday are representative of the composite score ratings that would be given by all possible consumers. It then follows that the composite score ratings given by the 60 sampled shoppers can be regarded as an approximately random sample that can be used to make statistical inferences about the population of all possible consumer composite score ratings.

## 统计代写|商业分析作业代写Statistical Modelling for Business代考|Selecting a Random Sample

0511715602854461399049191915
（见表 1.4(a)——这些数字用一个矩形括起来。）因为没有员工编号7156,4461,3990，或 4919（请记住，只有 2,136 名员工使用 500 分钟计划），我们忽略这些数字。这意味着前三个随机选择的员工是编号为 0511、0285 和 1915 的员工。继续这个过程，我们可以获得 100 名员工的整个随机样本。请注意，因为我们是在没有放回的情况下进行抽样，所以我们应该忽略之前从随机数表中选择的任何四位数字。

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## MATLAB代写

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