Question

ap stats lesson 4.2 homework name: claire coleman
1 a local amazon distribution center delivers products to three different states: illinois, michigan, and ohio. there were 14,500 purchases delivered from this distribution center on one day. amazon wants to send a customer - service survey to a sample of 100 customers who got one of these deliveries.
a. describe a method for selecting a simple random sample.
b. is it possible that there are no customers from michigan in the simple random sample of 100 customers? explain.
c. how could amazon ensure that customers from all three states are represented in the survey? explain.
2 a high - school principal for a small high school of 400 students wants to survey students about a new proposed requirement that athletes must maintain perfect school attendance in order to continue to participate in sports. she decides to select a stratified random sample of 40 students.
a. would it be better to stratify by grade level or by whether or not students are athletes? explain.
b. describe the process for selecting the stratified random sample.
3 a large fast - food restaurant has a total of 150 employees. there are 120 employees paid hourly and 30 employees on management salary. the owner of the restaurant wants to select a sample of 20 employees to attend a team meeting.
a. describe one potential issue with selecting an srs of 20 employees.
b. suppose the owner decides to take a stratified random sample by selecting an srs of 10 hourly wage employees and an srs of 10 management salary employees. are the employees within each group homogeneous or heterogeneous? explain.
c. the owner realizes that using equal - sized groups for the hourly wage employees and management salary employees is not representative of all employees. what sample sizes should be used for each group in order for the stratified random sample to be representative of all employees?

Explanation:

1a.

Label all 14,500 customers from 1 - 14500. Use a random - number generator or a table of random digits to select 100 unique numbers within the range of 1 - 14500. The customers corresponding to these numbers form the simple random sample.

1b.

Yes. In a simple random sample, every possible sample of size 100 from the 14,500 customers is equally likely. It is possible by chance that none of the selected 100 customers are from Michigan.

1c.

Amazon could use stratified sampling. First, determine the proportion of customers from each state (Illinois, Michigan, Ohio) out of the 14,500 total customers. Then, based on these proportions, select a number of customers from each state such that the sum of the customers from all three states is 100. For example, if the proportion of Michigan customers is $p_{MI}$, the number of Michigan customers in the sample would be $n_{MI}=100\times p_{MI}$.

2a.

It would be better to stratify by whether or not students are athletes. The proposed requirement is about athletes' school attendance, so the opinions of athletes and non - athletes are likely to differ more significantly than opinions based on grade level.

2b.

First, divide the 400 students into two strata: athletes and non - athletes. Determine the proportion of athletes and non - athletes in the population. Then, based on these proportions, calculate the number of students to be sampled from each stratum. For example, if there are 100 athletes and 300 non - athletes, and we want a sample of 40 students, the proportion of athletes is $\frac{100}{400}=0.25$. So the number of athletes in the sample is $40\times0.25 = 10$, and the number of non - athletes is $40 - 10=30$. Then, use a simple random sample within each stratum to select the appropriate number of students.

3a.

One potential issue with an SRS of 20 employees is that it may not accurately represent the two different types of employees (hourly - paid and management). There is a chance that the sample could be over - represented by one group and under - represented by the other. For example, it might end up with mostly hourly - paid employees or mostly management employees.

3b.

The employees within each group are heterogeneous. Among the hourly - paid employees, there are differences in job responsibilities, experience levels, etc. Similarly, among the management employees, there are differences in job titles, years of experience, and areas of responsibility.

3c.

The proportion of hourly - paid employees is $\frac{120}{150}=0.8$ and the proportion of management employees is $\frac{30}{150}=0.2$. For a sample of 20 employees, the number of hourly - paid employees in the sample should be $20\times0.8 = 16$, and the number of management employees should be $20\times0.2=4$.

Answer:

1a. Label all 14,500 customers from 1 - 14500. Use a random - number generator or table of random digits to select 100 unique numbers within 1 - 14500. The corresponding customers form the sample.
1b. Yes. In a simple random sample, every sample of size 100 is equally likely, and it's possible by chance to have no Michigan customers.
1c. Use stratified sampling. Determine the proportion of customers from each state, and select customers from each state based on these proportions so that the total is 100.
2a. It is better to stratify by whether or not students are athletes because the proposed requirement is about athletes' attendance, and opinions of athletes and non - athletes are likely to differ more than by grade level.
2b. Divide students into athletes and non - athletes strata. Calculate the proportion of each in the population, then the number of students to sample from each stratum. Use simple random sampling within each stratum to select the appropriate number of students.
3a. An SRS may not accurately represent the two types of employees (hourly - paid and management), and could over or under represent one group.
3b. The employees within each group are heterogeneous as there are differences in job responsibilities, experience levels, etc. within both hourly - paid and management groups.
3c. 16 hourly - paid employees and 4 management employees.