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You have been hired by the Regional Real Estate Company to help them analyze real estate data. One of the company’s Pacific region salespeople just returned to the office with a newly designed advertisement. It states that the average cost per square foot of his home sales is above the average cost per square foot in the Pacific region. He wants you to make sure he can make that statement before approving the use of the advertisement. The average cost per square foot of his home sales is $275. In order to test his claim, you collect a sample of 1,001 home sales for the Pacific region.

Sample Paper For Above instruction

The claim made by the Pacific region salesperson that his home sales have an average cost per square foot exceeding the regional average is a hypothesis that can be tested statistically. To evaluate this claim, a comprehensive analysis involving hypothesis formulation, data collection, and statistical testing is essential. The primary objective is to determine whether the sample data provide sufficient evidence to support the claim that the average cost per square foot of his home sales is greater than the regional average, which is assumed to be $275 based on the sample statistic.

Formulating the hypotheses correctly is crucial. The null hypothesis (H₀) posits that the true population mean cost per square foot of this salesperson’s home sales is less than or equal to the regional average, i.e., H₀: μ ≤ $275. Conversely, the alternative hypothesis (H₁) states that the true mean exceeds the regional average, H₁: μ > $275. This is a one-tailed test because the salesperson claims that the average cost per square foot is above the regional average.

Sampling data is collected from 1,001 home sales, which provides a robust sample size to increase the power of the statistical test. Assuming the sample mean (x̄) and sample standard deviation (s) are known or can be estimated, the next step involves calculating the test statistic. If the population standard deviation (σ) is known, a z-test is appropriate; otherwise, a t-test must be employed.

The test statistic for a z-test is computed as:

z = (x̄ - μ₀) / (σ / √n)

where x̄ is the sample mean, μ₀ is the hypothesized population mean ($275), σ is the population standard deviation, and n is the sample size (1001). If σ is unknown, the sample standard deviation (s) replaces σ, and a t-test is used with the formula:

t = (x̄ - μ₀) / (s / √n)

Once the test statistic is calculated, it is compared against critical values from the standard normal or t-distribution, depending on which test is used. The significance level (α), commonly set at 0.05, determines the threshold for statistical significance. If the test statistic exceeds the critical value, the null hypothesis is rejected in favor of the alternative, supporting the salesperson’s claim.

It is vital to also compute a confidence interval for the mean to understand the range of plausible values for the average cost per square foot. A 95% confidence interval provides a balance between precision and confidence, showing the range within which the true mean likely falls. If the lower bound of this interval exceeds $275, it further strengthens the case that the sales indeed have a higher average cost per square foot.

In conclusion, the statistical testing process involves collecting a large enough sample, accurately computing the test statistic, and evaluating it against critical values or p-values. If the data supports the claim at the chosen significance level, the salesperson can confidently use the advertisement asserting that his homes’ costs are above the regional average. This analysis ensures that claims made in marketing materials are substantiated with robust statistical evidence, maintaining the integrity of the real estate company's advertising practices.

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