Need some help filling out the work sheet. I have attached the descriptive Statistics summary from data from last week. The Sampling distributions real estate is the questions I need help with. I have also attached the excel spread sheet with the data information.
descriptive_statistics_summary.docx
sampling_distributions_real_estate_part2_.doc
descriptive_statistics__real_estate_data_part_1_excel.xlsx
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Descriptive Statistics- Real Estate Data Part 1
The frequency distribution of the listing prices ranged between $140,000 and $910,000.
The listing prices are classified using a class interval of $110,000. At the same time the home
listings are concentrated between $140,000 and $470,000 which there were 85 homes or 85
percent were in this range. The class midpoint of the first class was $195,000, (140,000 +
250,000) / 2 = 195,000. The largest concentration of price listings is in the $140,000 up to
$250,000 range. With that, the 48 homes in this class have a class midpoint of $195,000.
Frequency Distribution
Listing Price
$140,000 up to $250,000
250,000 up to 360,000
360,000 up to 470,000
470,000 up to 580,000
580,000 up to 690,000
690,000 up to 800,000
800,000 up tp 910,000
Relative
Frequency Frequency
48
0.480
23
0.230
14
0.140
5
0.050
2
0.020
5
0.050
3
0.030
Total
100
1.000
The frequency distribution of the houses having 3-6 bedrooms is presented in the table 1:
Table 1. Frequency distribution of the houses according to the number of bedrooms
Number of bedrooms
Absolute frequency
Relative frequency
6
3
0.03
5
18
0.18
4
41
0.41
3
38
0.38
According to the results presented in following histogram (figure 1), most of the houses
have 3-4 bedrooms, and only a few of them have 6 bedrooms. Taking this into account, the
bedrooms in the house do not follow a normal distribution since the average, median and mode
are concentrated on the smaller houses with less number of bedrooms.
Figure 1. Histogram plot representing the number of houses having 3-6 bedrooms
Figures 2 and 3 present a pie chart and a bar graph of the number of bedrooms in
the hundred houses considered in the present study.
Figure 2. Pie chart representing the number of houses having 3-6 bedrooms
Figure 3. Bar graph representing the distribution of houses with 3-6 bedrooms
Both charts have been prepared to enhance the visual aid by using bright colors and
avoiding the use of colors that cause problems with daltonism next to each other.
The statistical parameters of both listing prices and square footages are presented in table
Table 2. Statistical parameters of listing prices and square footages
Parameter
Listing price
Square footage
Mean
$323346
3163
Median
$259450
2679
Mode
$239900
2424
1st quartile
$192175
2324
3rd quartile
$395000
3823
Range
$750100
5433
Standard deviation
Variance
$181452
1202
$32924801312
1443890
According to Chebyshev’s theorem, no more than 1/k2 of the data can be outside the
range (mean ± k*st deviation), meaning that 75% of the values lie within the mean ± 2*st
deviation and 89% within the mean ± 3*st deviation. Taking this into account, we can calculate
if the data satisfy this rule by comparing the result obtained for the 3rd quartile (75% of the data)
with the value of mean ± 2*st deviation. Thus,
For the listing price:
Third quartile
395000
Mean + 2*st deviation
323346+2*181452 = 686250
For the square footage:
Third quartile
3823
Mean + 2*st deviation
3163+2*1202 = 5566
As we see, none of the distributions follows the Chebyshev theorem, since the upper limit
of the interval delimited by mean ± 2*st deviation range is higher than the third quartile in both
cases, meaning that more than 75% of the data lie within such interval. This can easily be
understood by considering the frequency table obtained previously, according to which most of
the houses were concentrated in the lower side of bedrooms. Hence, since the square footage is
closely related to the number of bedrooms in the house, and the listing price is dependent on the
square footage, we can expect that the frequency tables for the both the square footage and the
listing price will follow the same trend, such that the majority of the data will be concentrated on
the lower extreme.
On the other hand, according to the empirical rule, 68% of the data lie within the interval
defined by the mean ± standard deviation, 95% within the interval defined by mean ± 2*standard
deviation and 99% within the interval defined by mean ± 3*standard deviation. Table 3 presents
the comparison of the values obtained for the 68%, 95% and 99%, and the comparison between
these values with the intervals defined by mean ± standard deviation, mean ± 2*standard
deviation and mean ± 3*standard deviation, respectively.
Table 3. Verification of the empirical rule
Listing price
Square footage
68% percentile
347688
3370
Mean + st. deviation
504798
4364
95% percentile
799000
6103
Mean + 2*st. deviation
686250
5566
99% percentile
895000
7017
Mean + 3*st. deviation range
867702
6767
As can be observed, the data do not follow the empirical rule since there is a big
difference between the upper limit of the interval defined by “mean ± st. deviation range”, “mean
± 2*st. deviation range”, and “mean ± 3*st. deviation range” and the respective percentile values.
This is in accordance with our previous observation that the data did not follow a normal
distribution pattern.
Taking into account the obtained results, and considering the difference between the
obtained ranges, we can conclude that neither the Chebyshev’s theorem nor the Empirical rule
are able of describing the data corresponding to the listing price and the square footage of the
houses.
Sampling Distributions – Real Estate Part 2
Directions: Use the real estate data you used for your Week 2 learning team
assignment. Analyze the data and explain your answers.
1.
Review the data and for the purpose of this project please consider the 100 listing prices as a
population.
• Explain what your computed population mean and population standard deviation were.
2. Divide the 100 listing prices into 10 samples of n=10 each. Each of your 10 samples will tend to
be random if the first sample includes houses 1 through 10 on your spreadsheet, the second
sample consists of houses 11 through 20, and so on.
• Compute the mean of each of the 10 samples and list them:
3. Compute the mean of those 10 means.
• Explain how the mean of the means is equal, or not, to the population mean of the 100 listing
prices from above.
4. Compute the standard deviation of those 10 means and compare the standard deviation of the 10
means to the population standard deviation of all 100 listing prices.
• Explain why it is significantly higher, or lower, than the population standard deviation.
5. Explain how much more or less the standard deviation of sample means was than the population
standard deviation. According to the formula for standard deviation of sample means, it should be
far less. (That formula is σ = σ/√n = σ/√10 = σ/3.16 ) Does your computed σ agree with
the formula?
6. According to the Empirical Rule, what percentage of your sample means should be within 1
standard deviation of the population mean? Using your computed σ , do your sample means
seem to conform to the rule?
7. According to the Empirical Rule, what percentage of your sample means should be within 2
standard deviations of the population mean? Again, do your sample means seem to conform to
the rule?
8. You used the Empirical Rule because it really gives us more information (and because I asked
you to), but truthfully you should have used Chebyshev’s Theorem. Even though Chebyshev’s
doesn’t tell us much, why should you have used that one instead?
Address
237 Woodhill, Mankato
232 Woodhill, Mankato
112 S Mayfair Drive, Mankato
105 Larkspur, Mankato
121 Kestrel, Mankato
119 Red Oak, Mankato
133 Hidden Oaks, Mankato
101 Mount Curve, Mankato
20031 Hawkeye, Mankato
105 Sunburst, Mankato
148 Country Club, Mankato
2800 Monks, Mankato
129 Joshua, Mankato
101 Buckingham, Mankato
101 Manchester, Mankato
56601 200, Mankato
136 Pebble Creek, Mankato
817 Tanager, Mankato
100 Tanager, Mankato
1009 N 5th, Mankato
1946 Third, Mankato
10 Trail, Mankato
1001 Applewood, Mankato
102 Red Oak, Mankato
20455 598th, Mankato
120 Pollei, Mankato
124 Viking, Mankato
224 Sunset, Mankato
101 Jacob, Mankato
54557 Grouse, Mankato
112 Cedar, Mankato
317 Meadow Woods, Mankato
133 Timberwolf, Mankato
10 Hensonshire, Mankato
304 Woodhaven, Mankato
20186 589th, Mankato
304 Tanager, Mankato
370 Hensonshire, Mankato
105 Lynx, Mankato
421 Woodhaven, Mankato
108 Lynx, Mankato
205 Copperfield Drive, Mankato
19196 State Hwy 22, Mankato
204 Wickfield, Mankato
505 River Hills, Mankato
112 Center, Mankato
Listing Price
Square footage
$839,900
$799,900
$189,000
$895,000
$799,000
$675,000
$549,900
$459,900
$429,000
$415,000
$409,900
$399,900
$379,000
$349,899
$344,900
$269,900
$249,900
$239,900
$203,000
$174,500
$164,900
$895,000
$698,000
$599,000
$519,900
$510,000
$499,000
$495,000
$399,900
$395,000
$349,000
$339,000
$329,900
$329,500
$309,900
$299,900
$289,900
$275,000
$264,900
$262,000
$259,999
$258,900
$249,900
$239,900
$239,900
$239,900
Number of Bedrooms
6,920
6,120
2,603
6,271
6,206
4,853
4,138
5,780
3,837
3,416
4,217
5,088
3,840
4,089
4,124
3,328
2,612
2,448
2,810
2,712
2,256
7,018
4,202
5,704
4,692
4,142
3,250
3,879
4,774
5,014
3,300
4,434
3,782
3,082
2,984
2,912
2,548
2,344
2,772
2,640
2,472
2,290
3,378
2,424
2,456
4,066
6
6
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
212 Falcon, Mankato
229 Wadsworth, Mankato
653 Grayhawk Dr, Mankato
210 Dane, Mankato
204 Royal, Mankato
229 Cypress, Mankato
108 Trumpeter, Mankato
100 Olympic, Mankato
116 N Belmont, Mankato
200 Thro, Mankato
310 Fulton, Mankato
2712 E Main, Mankato
215 E Pleasant, Mankato
155 James, Mankato
1012 Monks, Mankato
208 Alta, Mankato
60418 200th, Mankato
54648 211, Mankato
2216 Shalom, Mankato
313 Dancing Waters, Mankato
117 Creekside, Mankato
100 Creekside, Mankato
309 Dancing Waters, Mankato
2009 Shalom, Mankato
116 Redtail, Mankato
117 Hope, Mankato
15 E Raven Rock, Mankato
125 Goldfinch, Mankato
304 Woodshire, Mankato
216 Muriefield, Mankato
300 Tranquility, Mankato
1028 Woodland Ave, Mankato
604 Tranquility, Mankato
305 Homestead, Mankato
105 Cole, Mankato
19975 S Hwy 22, Mankato
19982 Mill Rae, Mankato
144 Savannah, Mankato
132 Catalina, Mankato
104 W Welcome, Mankato
121 Jacob, Mankato
170 Terrace View East, Mankato
303 Floral, Mankato
424 Deldona, Mankato
212 Terrace, Mankato
808 W Pleasant, Mankato
505 W 8th, Mankato
$234,900
$234,000
$232,500
$225,000
$225,000
$224,900
$224,900
$223,900
$219,900
$219,900
$184,900
$182,700
$179,900
$175,500
$159,900
$156,600
$799,000
$699,000
$454,900
$419,900
$409,900
$395,000
$389,900
$379,900
$359,900
$334,900
$319,450
$309,900
$289,500
$285,900
$264,000
$254,900
$239,000
$226,500
$224,900
$219,900
$204,900
$199,000
$189,900
$184,900
$182,500
$179,900
$164,500
$159,900
$159,900
$155,000
$152,500
3,036
3,192
2,324
2,529
3,528
2,130
2,324
2,096
1,920
2,496
3,349
2,112
4,429
2,310
2,208
2,428
3,560
4,600
2,223
2,167
3,576
2,221
1,785
1,727
1,869
3,354
3,200
2,656
2,596
2,628
3,444
2,576
2,607
1,976
2,424
2,624
2,408
1,984
2,936
2,702
1,740
2,322
2,758
2,576
2,408
2,650
2,304
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
727 Belwood, Mankato
1424 Fair, Mankato
717 N 6th, Mankato
520 W Pleasant, Mankato
1016 N 4th, Mankato
928 N Broad, Mankato
324 N Broad, Mankato
Total
$150,000
$149,900
$149,900
$149,900
$146,000
$144,900
$144,900
$32,334,648
2,494
1,702
1,585
3,145
2,063
2,016
2,036
316,280
3
3
3
3
3
3
3
Number of bedrooms
3
4
5
6
Absolute frequency
38
41
18
3
Relative frequency
0.38
0.41
0.18
0.03
Histogram
45
40
Frequency
35
30
25
20
15
10
5
0
3
4
5
6
Number of bedrooms
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
3
6
18
5
41
4
38
3
0
10
20
30
40
50
Total number of Bedrooms Per House
Total Number of Bedrooms Per House
3
18
4
3
5
6
41
38
41
38
Frequency Distribution Summary
1. The home listing prices are in a range between $140,000 and $910,000.
2. The listing prices are classified using a class interval of $110,000.
3. The home listings are concentrated between $140,000 and $470,000.
There are 85 homes or 85% within this range.
4. The class midpoint of the first class is $195,000. (140,000 + 250,000)/2 = 195,000
5. The largest concentration of price listings is in the $140,000 up to $250,000 range.
There are 48 homes in this class with a class midpoint of $195,000.
Frequency Distribution
Listing Price
Frequency
$140,000 up to $250,000
48
250,000 up to 360,000
23
360,000 up to 470,000
14
470,000 up to 580,000
5
580,000 up to 690,000
2
690,000 up to 800,000
5
800,000 up tp 910,000
3
Total
100
Relative Frequency
0.480
0.230
0.140
0.050
0.020
0.050
0.030
1.000
Frequency Distribution Histogram
60
50
40
30
Frequency
20
10
0
$140,000 250,000
up to
up to
360,000
up to
470,000
up to
580,000
up to
690,000
up to
800,000
up tp
$140,000 250,000
up to
up to
$250,000 360,000
360,000
up to
470,000
470,000
up to
580,000
580,000
up to
690,000
690,000
up to
800,000
800,000
up tp
910,000
Parameter
Mean
Median
Mode
1st quartile
3rd quartile
Range
St. deviation
Variance
Percentile (68)
Percentile (95)
Percentile (99)
Mean+stdev
Mean+2stdev
Mean+3stdev
Listing price
$323,346
$259,450
$239,900
$192,175
$395,000
$750,100
$181,452
$32,924,801,312
$347,688
$799,000
$895,000
$504,798
$686,250
$867,702
Square footage
3,163
2,679
2,424
2,324
3,823
5,433
1,202
1,443,890
3370
6103
7017
4,364
5,566
6,768
Quartiles for Price Listings
Minimum Value
1st Quartile
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Median Value
3rd Quartile
Maximum value
895000
395000
259449.5
196725
144900
1
Quartiles For Square Footage
7018
3795.75
2679
2324
1585
1
MINIMUM VALUE
1ST QUARTILE
Median Value
3rd Quartile
MAXIMUM VALUE
…
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