Question

The B&K Real Estate Company sells homes and is currently serving the Southeast region. It has recently expanded to cover the Northeast states. The B&K realtors are excited to now cover the entire East Coast and are working to prepare their southern agents to expand their reach to the Northeast.

B&K has hired your company to analyze the Northeast home listing prices in order to give information to their agents about the mean listing price at 95% confidence. Your company offers three analysis packages: one based on a sample size of 100 listings, one based on 1,000 listings, and another based on a sample size of 4,000 listings. Because there is an additional cost for data collection, your company charges more for the package with 4,000 listings than for the package with 100 listings.

Bronze Package - Sample size of 100 listings:

95% confidence interval for the mean of the Northeast house listing price has a margin of error of $24,500
Cost for service to B&K: $2,000
Silver Package - Sample size of 1,000 listings:

95% confidence interval for the mean of the Northeast house listing price has a margin of error of $7,750
Cost for service to B&K: $10,000
Gold Package - Sample size of 4,000 listings:

95% confidence interval for the mean of the Northeast house listing price has a margin of error of $3,900
Cost for service to B&K: $25,000
The B&K management team does not understand the tradeoff between confidence level, sample size, and margin of error. B&K would like you to come back with your recommendation of the sample size that would provide the sales agents with the best understanding of northeast home prices at the lowest cost for service to B&K.

In other words, which option is preferable?

Spending more on data collection and having a smaller margin of error
Spending less on data collection and having a larger margin of error
Choosing an option somewhere in the middle
For your initial post:

Formulate a recommendation and write a confidence statement in the context of this scenario. For the purposes of writing your confidence statement, assume the sample mean house listing price is $310,000 for all packages. "I am [#] % confident the true mean . . . [in context]."
Explain the factors that went into your recommendation, including a discussion of the margin of error
For your response posts to your peers, choose two different confidence intervals for your responses. Do you think the agents would prefer a different confidence interval than their management? What advantages and disadvantages would there be in having different confidence intervals for the agents? Explain your thought process and reasoning in your response.

Answer 1

Explanation:

Based on the provided information, I would recommend B&K Real Estate Company to choose the Silver Package with a sample size of 1,000 listings. This option strikes a balance between cost and accuracy, providing a reasonable margin of error while minimizing the cost for the service.

Confidence Statement:

"I am 95% confident that the true mean of the Northeast house listing prices falls within the estimated range, based on a sample size of 1,000 listings."

Factors Considered:

Cost: The cost for the Silver Package is $10,000, which is lower than the Gold Package but higher than the Bronze Package. B&K can achieve a reasonably accurate estimate of the mean listing price at a lower cost compared to the Gold Package.

Margin of Error: The margin of error for the Silver Package is $7,750. While it is larger than the Gold Package, it is significantly smaller than the Bronze Package. The margin of error represents the range within which the true mean of listing prices is likely to fall. With a smaller margin of error, the estimated mean is more precise, providing a better understanding of the Northeast home prices compared to the Bronze Package.

Sample Size: The sample size of 1,000 listings in the Silver Package is larger than the Bronze Package but smaller than the Gold Package. A larger sample size generally leads to a more accurate estimate of the population mean. Although the sample size in the Gold Package is larger, the incremental gain in accuracy is not significant enough to justify the higher cost.

Response to Peers:

Response 1:

If the agents prefer a higher confidence interval, such as 99%, compared to the management's 95% confidence interval, it would require a larger sample size to achieve the desired accuracy. This could result in increased costs for data collection. However, a higher confidence interval provides greater confidence in the estimated mean listing price, giving the agents more assurance when making decisions. The advantage is a higher level of confidence, but the disadvantage is the increased cost associated with a larger sample size.

Response 2:

If the agents prefer a lower confidence interval, such as 90%, compared to the management's 95% confidence interval, it would require a smaller sample size to achieve the desired accuracy. This could reduce the cost for data collection. However, a lower confidence interval means a wider margin of error and less certainty in the estimated mean listing price. The advantage is the cost savings, but the disadvantage is a lower level of confidence in the estimate, potentially leading to more uncertainty for the agents when making decisions