Question

For complete credit show all of your calculations, list your assumptions and formulas, draw relevant tables and plots all on handwritten pages in your notebook.

If there are repetitious calculations you only need to show an illustrative calculation.

Submit pictures of your handwritten pages no later than Wednesday, March 4th by 1800 hours ending.

Use this sample of house prices and lotsizes in the Pelham Bay neighborhood of the Bronx from 2018-2019 to answer the questions below.

price	lotsize
490000	2503
512000	2483
345000	2500
508670	2900
550000	2513
300000	2513
995000	4950
920000	3135
470000	2375
450000	2375

1. What kind of data is this?
2. For one of the series, and for 3 equally spaced non-overlapping intervals what in the interval size, the beginning, end and midpoint of each interval, the frequency, relative frequency, and cumulative relative frequency of each interval? Sketch the histogram and ogive.
3. What is the maximum, minimum, first quartile, median, third quartile, IQR, range, arithmetic mean, and standard deviation of the ungrouped data?
4. Sketch a boxplot with Tukey's fences. Are there any outliers?.
5. Calculate the weighted average of price and lotsize. Why are these the same or different from the median and arithmetic mean?
6. What is the correlation between price and lotsize?
7. For the model Y = a + bX + e, where Y = price, X = lotsize, e = deviation of Y from the conditional mean a + bX, what are the best estimators of the slope and the Y-intercept? Sketch the line with these parameters that runs through the scatterplot of Y and X. Be sure to draw the error bars from the data points to the line.

Answer 1

1)

Both the variables are continuous

2)

I am using lotsize

Steps for Histogram

• Sort the dataset in ascending order.
• Create 3 Intervals of 1000 width starting from 2000 and count the frequency for each bin
• Plot a histogram as shown below

Lower Limit	Upper Limit	Mid point	Interval	Freq	Relative Freq	Cumulative Freq
2000	2999	2499.5	2000-2999	8	0.8	8
3000	3999	3499.5	3000-3999	1	0.1	9
4000	4999	4499.5	4000-4999	1	0.1	10

Steps to create a Ogive (Cumulative frequency Graph)

• Calculate Cumulative Frequency.
• Select Data and click on Insert option
• Select the line chart option

3)

From the data, we calculate the mean and standard deviation of the two groups.

Mean (X bar) = Sum of Values /n

and

	price	lotsize
Maximum	995000	4950
Minimum	300000	2375
First Quartile (Q1)	455000	2487.25
Median	499335	2508
Third Quartile (Q3)	540500	2803.25
IQR (Q3-Q1)	85500	316
Range (Max-Min)	695000	2575
Mean	554067	2824.7
Std Dev	226675.7	784.8078

4)

As evident from the above boxplots, both price and lotsize have outliers.

5)

Weighted Average is same the mean as calculated in part 3

6)

	lotsize (x)	price (y)	xy	x2	y2
	2,503	4,90,000	12264,70,000	62,65,009	2401000,00,000
	2,483	5,12,000	12712,96,000	61,65,289	2621440,00,000
	2,500	3,45,000	8625,00,000	62,50,000	1190250,00,000
	2,900	5,08,670	14751,43,000	84,10,000	2587451,68,900
	2,513	5,50,000	13821,50,000	63,15,169	3025000,00,000
	2,513	3,00,000	7539,00,000	63,15,169	900000,00,000
	4,950	9,95,000	49252,50,000	245,02,500	9900250,00,000
	3,135	9,20,000	28842,00,000	98,28,225	8464000,00,000
	2,375	4,70,000	11162,50,000	56,40,625	2209000,00,000
	2,375	4,50,000	10687,50,000	56,40,625	2025000,00,000
Total	28,247	55,40,670	169659,09,000	853,32,611	35323391,68,900

Using the above formula and values, we get

Correlation coefficient (r) = 0.821

The magnitude tells us that there is strong correlation between the two variables and the sign of r tells us that there is direct relationship between the two ie as lotsize increases, price also increases and vice versa.

7)

Let the regression equation be: Y = a + bX

Where

Slope(b) = {n*∑XY - ∑X *∑Y}/{n*∑X² – (∑X)² } = 237.26

and a = ∑Y/n – b*∑X/n = -1,16,107.41

Hence, Price = 237.26 * Lotsize - 1,16,107.41