Hello, could someone explain which formula or equation I should use to solve this? I'm finding myself a bit stumped by the question as I don't understand what confidence intervals are or how they relate to margin errors. Don't give me the exact answer, I'd rather learn how to do this on my own. Please just give me a guide for how to solve this, including an explanation on confidence intervals and margin of errors because I just don't understand them very well.

Here is the 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