Calculating the Proportion of Digital Camera Users

Oct 13, 2025 by wbrum.org 68 views

Introduction

Cluster sampling is a type of sampling method used in statistics to select a representative sample from a population. In this method, the population is divided into clusters, and a random sample of clusters is selected. The individuals within the selected clusters are then surveyed. Cluster sampling is often used when the population is large and difficult to access, or when the data is expensive to collect.

The Problem

Suppose cluster sampling were being used to survey digital camera users, who amount to $77\%$ of the population of the United States. The table below shows the number of digital camera users in each city, along with the total population of each city.

City	Total Population	Digital Camera Users
New York	8,420,527	6,475,111
Los Angeles	3,999,759	3,085,111
Chicago	2,670,504	2,051,111
Houston	2,355,386	1,811,111
Phoenix	1,732,486	1,333,111
Philadelphia	1,567,872	1,205,111
San Antonio	1,567,118	1,205,111
San Diego	1,425,976	1,095,111
Dallas	1,344,378	1,033,111
San Jose	1,013,240	781,111

The Question

Which city would be considered the best to survey digital camera users using cluster sampling?

The Solution

To determine which city would be considered the best to survey digital camera users using cluster sampling, we need to calculate the proportion of digital camera users in each city. We can then compare these proportions to determine which city has the highest proportion of digital camera users.

Calculating the Proportion of Digital Camera Users

To calculate the proportion of digital camera users in each city, we can use the following formula:

Proportion = (Number of Digital Camera Users / Total Population) x 100

We can apply this formula to each city in the table:

City	Proportion of Digital Camera Users
New York	(6,475,111 / 8,420,527) x 100 = 77.1%
Los Angeles	(3,085,111 / 3,999,759) x 100 = 77.1%
Chicago	(2,051,111 / 2,670,504) x 100 = 77.3%
Houston	(1,811,111 / 2,355,386) x 100 = 76.8%
Phoenix	(1,333,111 / 1,732,486) x 100 = 76.7%
Philadelphia	(1,205,111 / 1,567,872) x 100 = 76.9%
San Antonio	(1,205,111 / 1,567,118) x 100 = 76.9%
San Diego	(1,095,111 / 1,425,976) x 100 = 76.8%
Dallas	(1,033,111 / 1,344,378) x 100 = 76.7%
San Jose	(781,111 / 1,013,240) x 100 = 77.2%

Comparing the Proportions

Now that we have calculated the proportion of digital camera users in each city, we can compare these proportions to determine which city has the highest proportion of digital camera users.

From the table above, we can see that Chicago has the highest proportion of digital camera users, with a proportion of 77.3%. This means that Chicago has the highest percentage of digital camera users among all the cities listed in the table.

Conclusion

In conclusion, based on the table provided, Chicago would be considered the best city to survey digital camera users using cluster sampling. This is because Chicago has the highest proportion of digital camera users, with a proportion of 77.3%.

Limitations of Cluster Sampling

Cluster sampling is a useful method for selecting a representative sample from a population, but it has some limitations. One of the main limitations of cluster sampling is that it can be difficult to select a representative sample of clusters. Additionally, cluster sampling can be expensive and time-consuming, especially if the population is large and difficult to access.

Future Research Directions

Future research directions in cluster sampling could include developing new methods for selecting a representative sample of clusters, as well as improving the efficiency and cost-effectiveness of cluster sampling. Additionally, researchers could investigate the use of cluster sampling in different fields, such as medicine and social sciences.

References

Cochran, W. G. (1977). Sampling techniques. John Wiley & Sons.
Kish, L. (1965). Survey sampling. John Wiley & Sons.
Levy, P. S., & Lemeshow, S. (2009). Sampling of populations: Methods for analysis. John Wiley & Sons.

Appendix

The following table shows the number of digital camera users in each city, along with the total population of each city.

City	Total Population	Digital Camera Users
New York	8,420,527	6,475,111
Los Angeles	3,999,759	3,085,111
Chicago	2,670,504	2,051,111
Houston	2,355,386	1,811,111
Phoenix	1,732,486	1,333,111
Philadelphia	1,567,872	1,205,111
San Antonio	1,567,118	1,205,111
San Diego	1,425,976	1,095,111
Dallas	1,344,378	1,033,111
San Jose	1,013,240	781,111

Q: What is cluster sampling?

A: Cluster sampling is a type of sampling method used in statistics to select a representative sample from a population. In this method, the population is divided into clusters, and a random sample of clusters is selected. The individuals within the selected clusters are then surveyed.

Q: What are the advantages of cluster sampling?

A: The advantages of cluster sampling include:

It is a cost-effective method, especially when the population is large and difficult to access.
It is a simple method to implement, especially when the population is already divided into clusters.
It can be used to select a representative sample from a population that is difficult to access.

Q: What are the disadvantages of cluster sampling?

A: The disadvantages of cluster sampling include:

It can be difficult to select a representative sample of clusters.
It can be expensive and time-consuming, especially if the population is large and difficult to access.
It can lead to biased results if the clusters are not representative of the population.

Q: How do you calculate the proportion of digital camera users in a city using cluster sampling?

A: To calculate the proportion of digital camera users in a city using cluster sampling, you can use the following formula:

Proportion = (Number of Digital Camera Users / Total Population) x 100

Q: What is the difference between cluster sampling and simple random sampling?

A: The main difference between cluster sampling and simple random sampling is that cluster sampling involves selecting a random sample of clusters, while simple random sampling involves selecting a random sample of individuals from the population.

Q: Can cluster sampling be used in medicine and social sciences?

A: Yes, cluster sampling can be used in medicine and social sciences. For example, it can be used to select a representative sample of patients for a clinical trial, or to select a representative sample of individuals for a survey on social issues.

Q: How do you ensure that the clusters selected are representative of the population?

A: To ensure that the clusters selected are representative of the population, you can use the following methods:

Use a stratified sampling method to select clusters from different strata of the population.
Use a random sampling method to select clusters from the population.
Use a combination of both stratified and random sampling methods to select clusters.

Q: What are some common applications of cluster sampling?

A: Some common applications of cluster sampling include:

Selecting a representative sample of patients for a clinical trial.
Selecting a representative sample of individuals for a survey on social issues.
Selecting a representative sample of businesses for a market research study.

Q: Can cluster sampling be used in conjunction with other sampling methods?

A: Yes, cluster sampling can be used in conjunction with other sampling methods, such as stratified sampling and simple random sampling. This is known as a multi-stage sampling method.

Q: What are some common challenges associated with cluster sampling?

A: Some common challenges associated with cluster sampling include:

Selecting a representative sample of clusters.
Ensuring that the clusters selected are representative of the population.
Dealing with non-response and missing data.

Q: How do you deal with non-response and missing data in cluster sampling?

A: To deal with non-response and missing data in cluster sampling, you can use the following methods:

Use a imputation method to replace missing data with estimated values.
Use a weighting method to adjust for non-response.
Use a combination of both imputation and weighting methods to deal with non-response and missing data.

Q: What are some common software packages used for cluster sampling?

A: Some common software packages used for cluster sampling include:

R
SAS
SPSS
Stata

Q: Can cluster sampling be used in conjunction with other statistical methods?

A: Yes, cluster sampling can be used in conjunction with other statistical methods, such as regression analysis and hypothesis testing. This is known as a multi-method approach.