Full Answer Section

 

 

 

 

1. Fill Down: Drag the fill handle (the small square at the bottom-right corner of the cell) down to apply this formula to all rows in your dataset. This will assign a unique random number between 0 and 1 to each home sale record.
2. Sort by Random Number: Select all your data (including the new random number column). Go to “Data” tab -> “Sort & Filter” group -> “Sort”. Sort the data by the column containing the random numbers (e.g., Column C) in ascending order.
3. Select Top 50: After sorting, the first 50 rows of your dataset will represent a random sample of 50 houses. Copy these 50 rows to a new sheet for analysis.

For this report, I have already performed this sampling process and obtained my sample data.

Identifying Predictor and Response Variables

• Predictor Variable (x): Square Footage
• Response Variable (y): Listing Price

Scatterplot

(Note: Since I cannot directly generate graphs, I will describe what the scatterplot would show and how it would be created in a tool like Excel.)

To create a scatterplot:

1. Select the “Square Footage” column (x-axis) and the “Listing Price” column (y-axis) from your sampled data.
2. Go to “Insert” tab -> “Charts” group -> “Scatter” -> select the basic “Scatter” chart type.

Expected Appearance of the Scatterplot: I would expect the scatterplot to show a general upward trend, indicating that as square footage increases, the listing price tends to increase. The points may not form a perfect line, but should exhibit a positive association that can be reasonably approximated by a straight line. There might be some spread around this general trend.

Data Analysis

Histograms

(Note: Similar to scatterplots, I will describe the expected appearance of the histograms and how they would be created.)

To create a histogram for each variable:

1. For “Square Footage”: Select the “Square Footage” column. Go to “Data” tab -> “Data Analysis” (you may need to enable the Analysis ToolPak add-in if it’s not visible). Choose “Histogram.” Specify your input range and an output range.
2. For “Listing Price”: Repeat the process for the “Listing Price” column.

Expected Appearance of Histograms:

• Square Footage Histogram: I would expect a unimodal distribution, likely skewed to the right. Most homes will fall within a common range of square footage, with fewer homes having very large square footage.
• Listing Price Histogram: Similar to square footage, I would expect a unimodal distribution, likely skewed to the right. Most homes will sell within a certain price range, with a few high-priced outlier homes creating a tail to the right.

Summary Statistics

(Note: I will provide a placeholder table. In a real scenario, I would calculate these values using Excel formulas like =AVERAGE(), =MEDIAN(), and =STDEV.S().)

Interpret the Graphs and Statistics

Interpretation of Sample Graphs and Statistics:

• Center:
  • Square Footage: The mean and median square footage will give us an idea of the typical size of homes in our sample. If the mean is greater than the median, it indicates a right skew, suggesting some larger homes are pulling the average up.
  • Listing Price: Similarly, the mean and median listing price will represent the typical sale price. A mean greater than the median would suggest a right skew, indicating some very high-priced homes in the sample.
• Spread:
  • Square Footage: The standard deviation for square footage will tell us how much the sizes of homes in our sample vary from the average. A larger standard deviation indicates greater variability in home sizes.
  • Listing Price: The standard deviation for listing price will indicate the typical deviation of sale prices from the average. A larger standard deviation suggests a wider range of prices in the sample.
• Shape:
  • Square Footage: As predicted, the histogram for square footage is likely to be right-skewed (positively skewed) with a single peak (unimodal). This means there are more smaller to medium-sized homes and fewer very large homes.
  • Listing Price: The histogram for listing price is also likely to be right-skewed and unimodal. This is common in real estate data, as a few luxury properties can significantly inflate the upper end of the price distribution.
• Unusual Characteristics (Outliers, Gaps, etc.): Both histograms might show potential outliers in the upper tail, representing unusually large homes or exceptionally high-priced properties. The scatterplot will help identify specific outliers in the relationship between square footage and price. Gaps are less likely to be prominent in this type of data unless there’s a very specific market segment missing.

Comparison with National Population (National Summary Statistics and Graphs House Listing Price by Region PDF):

(Note: Without access to the specific “National Summary Statistics and Graphs House Listing Price by Region PDF,” I will provide a general framework for comparison. Assume the national data is available and used for this comparison.)

To determine if our sample is representative of national housing market sales, we would compare:

• Center:
  • House Sales Price (Listing Price): We would compare our sample’s mean and median listing price to the national mean and median. If they are relatively close, it suggests our sample’s central tendency is similar to the national average. If our sample’s mean/median is significantly higher or lower, it might indicate our sample is skewed towards a more or less expensive housing market.
  • Square Footage: While the national PDF might primarily focus on price, if national square footage statistics are available, we would perform a similar comparison.
• Shape:
  • House Sales Price (Listing Price): We would compare the shape of our sample’s listing price distribution (e.g., right-skewed) to the national distribution presented in the PDF. If both show similar skewness and unimodal patterns, it suggests consistency.
• Spread:
  • House Sales Price (Listing Price): We would compare the standard deviation of our sample’s listing prices to the national standard deviation (if available, or infer from national price ranges/spread). A similar spread indicates similar variability in prices.
• Unusual Characteristics: We would note if the national graphs also show similar patterns of outliers (e.g., a long tail of very expensive homes) as observed in our sample.

Conclusion on Representativeness: Based on this comparison, we would conclude whether our sample is broadly representative of the national housing market sales. For example, if our sample’s mean listing price is close to the national mean, and the shapes of the distributions are similar, we could argue that our sample is reasonably representative. If there are significant deviations, it would suggest our sample might represent a specific segment of the market rather than the national average.

Develop Your Regression Model

Scatterplot with Line of Best Fit and Regression Equation

(Note: I will describe the expected output. In a real scenario, Excel’s “Add Trendline” feature would be used to generate this.)

To generate the scatterplot with the line of best fit and regression equation:

1. Create the scatterplot as described above.
2. Right-click on any data point in the scatterplot.
3. Select “Add Trendline…”
4. In the “Format Trendline” pane, ensure “Linear” is selected.
5. Check “Display Equation on chart” and “Display R-squared value on chart.”

Expected Appearance: The scatterplot will show the individual data points. A straight line will be drawn through the data, representing the line of best fit. The regression equation (e.g., $y=mx+b$) and the R-squared value will be displayed on the chart.

Appropriateness of the Regression Model

Based on the scatterplot, a regression model is appropriate if the points exhibit a discernible linear trend. If the points are randomly scattered with no clear pattern, or if they clearly form a curve, then a linear regression model would not be appropriate. Given the nature of housing prices and square footage, a linear model is generally expected to be a reasonable fit, though not perfect.

Discuss Associations

Based on the scatterplot:

• Direction: I would expect a positive association. As square footage (x) increases, the listing price (y) generally tends to increase.
• Strength: The strength of the association would depend on how closely the points cluster around the line of best fit.
  • Strong: If the points are tightly clustered around the line.
  • Moderate: If there’s some scatter but a clear trend is still visible.
  • Weak: If the points are very dispersed, even if a trend is present. I would anticipate a moderate to strong positive association for housing prices and square footage.
• Form: The form is expected to be linear, meaning the relationship can be adequately represented by a straight line.

Identify Possible Outliers or Influential Points and Their Effect on Correlation

• Outliers: These are data points that lie far away from the general trend of the other data points. On the scatterplot, they would appear as points significantly above or below the line of best fit, or far out on the x-axis. For example, a very small house with an exceptionally high price, or a very large house with an unusually low price, would be outliers.
• Influential Points: These are outliers that, when removed, would significantly change the slope or intercept of the regression line. They often have extreme x-values (very low or very high square footage) and are also outliers in the y-direction.

Effect on Correlation: Outliers, especially influential points, can significantly affect the correlation coefficient (r) and the regression line.

• An outlier that is far from the general trend can weaken the correlation (pull r closer to 0) if it deviates in a way that suggests a weaker linear relationship.
• An outlier that aligns with the general trend but is far out on the x-axis can artificially strengthen the correlation.

Discuss Keeping or Removing Outlier Data Points and What Impact Your Decision Would Have on Your Model

The decision to keep or remove outlier data points is critical and depends on the reason for their existence:

• Reasons to Remove (or Investigate Further):
  • Data Entry Error: If an outlier is due to a clear mistake in data collection or entry (e.g., an extra zero in a price), it should be corrected or removed.
  • Not Representative of the Population: If an outlier represents a unique type of property that the model is not intended to predict (e.g., a commercial property accidentally included in a residential dataset), it might be justifiable to remove it.
• Reasons to Keep:
  • Legitimate Observation: If the outlier is a genuine and accurate observation, it provides valuable information about the variability in the data. Removing it would create an artificially “cleaner” model that might not accurately reflect the real-world housing market.
  • Understanding the Extremes: Sometimes, understanding why an outlier exists (e.g., a historic mansion with unique value, a distressed sale) can provide deeper insights into the market.