Suppose the average size (in square feet) of homes in a certain region is 1747. Assume the variable is normally distributed with standard deviation 220. If a person in the region was intending to build a 1890 square foot home but downsized and built a 1274 square foot home, what was the change in percentile rank from the original plan to the actual home that was built?

To find the change in percentile rank, we need to calculate the percentiles for both the original plan and the actual home that was built.

First, let's find the percentile rank for the original plan of 1890 square feet.

1. Calculate the z-score:
z = (X - μ) / σ
where X is the original plan size, μ is the mean, and σ is the standard deviation.

z = (1890 - 1747) / 220
z ≈ 0.650

2. Use a z-table or a statistical software to find the percentile corresponding to the z-score. The percentile represents the percentage of values below that particular observation in a normal distribution.

From the z-table or using a statistical software, the percentile for z = 0.650 is approximately 74.68%.

Now, let's find the percentile rank for the actual home that was built, which has a size of 1274 square feet.

1. Calculate the z-score:
z = (X - μ) / σ

z = (1274 - 1747) / 220
z ≈ -2.154

2. Use a z-table or a statistical software to find the percentile corresponding to the z-score.

From the z-table or using a statistical software, the percentile for z = -2.154 is approximately 1.63%.

Finally, calculate the change in percentile rank:

Change in percentile rank = Percentile rank (original plan) - Percentile rank (actual home)

Change in percentile rank = 74.68% - 1.63%
Change in percentile rank ≈ 73.05%

Therefore, the change in percentile rank from the original plan to the actual home that was built is approximately 73.05%.