Statistic Lab

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1. First enter the Real estate data (text book data set 1) into SPSS and then

select a sample of size, n = last two digits of your ID and answer the

exercises.

I. Select an appropriate class interval and organize the “Selling price” into

a frequency distribution.

II. Compute the Mean, Median, Mode, Standard Deviation, Variance,

Quartiles, 9th Decile, 10th Percentile and Range of “Selling price” from

the raw data of your sample and interpret.

III. Develop a histogram (Using question “1”) for the variable “selling

cost”.

IV. Develop a Pie chart and a Bar diagram for the variable “Township”.

V. Develop a Box plot for the variable “Distance”.

What information can you give from these plots?

Note: Comment on all your findings, charts and diagrams.

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Answer to the question No.1II. Computation of the Mean, Median, Mode, Standard Deviation, Variance, Quartiles, 9th

Deciles, 10th Percentile and Range of “Selling price” from the raw data of sample, n=64 :

From SPSS output, we get,Selling Price in TK. 000 (Thousand)

N Valid 64

Missing 0

Mean 2192.659

Median 2090.500

Mode 1880.3(a)

Std. Deviation 462.2481

Variance 213673.30

4

Percentiles 10 1707.000

20 1807.000

25 1880.300

30 1910.550

40 2050.100

50 2090.500

60 2220.100

70 2415.350

75 2468.050

80 2540.300

90 2930.350

Interpretation

Mean: On average the selling price of homes sold in Denver, Colorado is about TK. 2192.659

(Thousands).

Median: 50% of the selling price sold in Denver, Colorado is less than TK. 2090.500

(Thousands) and 50% of the selling price is above TK. 2090.500 (Thousands).

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Mode: The maximum number of selling price of homes is TK1880.3 (Thousands) which appears

more than any other selling price. In other words the homes has been sold maximum in the price

of TK. 1880.3 (Thousands).

Std. Deviation: The actual amount of selling price of homes on average differs/varies from the

mean selling price TK2192.659thousands by TK462.2481 (Thousands).

Variance: The average variation of the selling price is Tk. 213673.304 (thousands).

Range: The range of selling price of homes is TK. 2199.4 (Thousands) where highest amount of

selling price of homes is TK.3450.3 (Thousands) and lowest amount of selling price of homes is

TK.1250.9 (Thousands).

Quartiles: The selling price of first 25% homes is equal or less than TK. 1880.300 thousands

and 75% homes is higher than TK. 1880.300 thousands that is presented as the first quartile, =

TK. 1880.300 thousands.

The median amount of selling price of homes is given by second quartile, =TK. 2090.500

(Thousands) i.e. the selling price of the 50% homes is less than TK2090.500 (Thousands) and the

selling price of the other 50% homes is higher than TK2090.500 (Thousands) .

The Selling price of the 75% homes is equal or less than TK. 2468.050 thousands and 25%

homes is higher than TK. 2468.050 thousands, denoted as third quartile, =TK. 2468.050

thousands.

9th Deciles: The selling price of the 90% homes sold in Denver, Colorado is equal or less than

TK. 2930.350 (Thousands) and 10% homes are above TK. 2930.350 (Thousands) .

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10th Percentile: The selling price of the first 10 percent homes is equal or less than Tk.

1707.000 thousands i.e. given by or =Tk. 1707.000 and 90 % homes are above it.IV.

Pie chart for the variable “Township”

Figure: Pie chart for the variable “Township”

12.5%

25.0%

25.0%

17.19%

20.31%

BananiDhanmondiDOHSUttaraGulshan

Pie chart for the variable “Township”

Comment: From the pie chart we see that, 25.0% of the Township is in Dhanmondi, 25.0% of

the Township is in DOHS, 20.31% of the Township is in Gulsan, 17.19% of the Township is in

Uttara, and 12.5% of the Township is in Banani. We also see that, the largest percentage of

Township involves with Dhanmondi, DOHS and the lowest percentage of Township involves

with Banani.

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Figure: Bar chart for the variable “Township”

BananiDhanmondiDOHSUttaraGulshan

Township

20

15

10

5

0

Freq

uenc

y

12.5%

25.0%25.0%

17.19%

20.31%

Comment: From the Bar chart we see that, 25% of the Township is in Dhanmondi, 25% of the

Township is in DOHS, 20.31% of the Township is in Gulsan, 17.19% of the Township is in

Uttara, and 1205% of the Township is in Banani. We also see that, the largest percentage of

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Township involves with Dhanmondi, DOHS and the lowest percentage of Township involves

with Banani.

V. Box plot for the variable “Distance”

Figure: Box plot of “Distance”

Distance

30.0

25.0

20.0

15.0

10.0

5.0

__

Comment: There is no outlier. The median distance is 15 and about 50 percent distance is

between 11 to19.

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2. For the variable “Selling price”

I. Determine the coefficient of skewness. Is the distribution positively or negatively

skewed?

II. Develop a Box plot. Are there any outliers

Answer to the question No.2I. From SPSS output, we get,

Statistics

Selling Price in TK. 000 (Thousand)

N Valid 64

Missing 0

Skewness .665

Std. Error of Skewness .299

Kurtosis .280

Std. Error of Kurtosis .590

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3500.03000.02500.02000.01500.01000.0


12

10

8

6

4

2

0

Freq

uenc

y

Mean = 2192.659Std. Dev. = 462.2481N = 64


__

Comment: The distribution is positively skewed. Since mean is the largest and mode is the

smallest average.

ii. Box plot for the variable “Selling price”

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Figure: Box plot of “Selling price”


3500.0

3000.0

2500.0

2000.0

1500.0

1000.0

17

__

Comment: There is one outlier that is 17.

3. Refer to the Real Estate data, which reports information on the homes sold in Denver,

Colorado, last year. [Chapter-9]

I. Develop a 95 percent confidence interval for the mean selling price of the homes and interpret.

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II. Develop a 95 percent confidence interval for the mean distance the home is from the center of

the city and interpret.

III. Develop a 95 percent confidence interval for the proportion of homes with an attached garage

and interpret.

Answer to the Question No.3

I. For 95 percent confidence interval for the mean selling price of the homes

Using SPSS we get the following output as-

One-Sample Statistics

One-Sample Test

One-Sample Test

Test Value = 0

t df Sig. (2-tailed)Mean

Difference

95% Confidence Interval of the Difference

Lower UpperSelling Price in TK. 000 (Thousand) 38.337 63 .000 2169.0381 2055.976 2282.100

So the 95% Confidence Interval for the mean selling price of the homes is ($2055.976,

$2282.100).

From the above table we have, Mean = 2169.038& Std. deviation 452.6248

The value of t at 95 % level of confidence & 63 df is 1.998 We know the confidence interval,

CI = ± t = 2192.659± 115.446 = ($1977.193, $2208.126) which is the 95% Confidence

Interval for the mean selling price of the homes.

N Mean Std. DeviationStd. Error

MeanSelling Price in TK. 000 (Thousand) 64 2169.038 452.6248 56.5781

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Interpretation:

Probability ($1977.193≤ μ ≤ , $2208.126) = 0.95

The mean selling price of the homes ranges from $1977.193 to $2208.126 we are 95% confident

about it.

In other words, this interval ($2077.193, $2308.126) contains the population mean with

probability 0.95. If we take a sample of Selling Price 100 times this interval will contain the

population 95 times.

II. 95 percent confidence interval for the mean distance the home is from the center of the

city:

Using SPSS, we get the following output as-

One-Sample Statistics

N Mean Std. Deviation

Std. Error

Mean

Distance 64 14.91 5.291 .661

One-Sample Test

Test Value = 100

t df Sig. (2-tailed)

Mean

Difference

95% Confidence Interval

of the Difference

Lower Upper

Distance -128.670 63 .000 -85.094 -86.42 -83.77

95 percent confidence interval for the mean distance the home is from the center of the city (-

86.42, -83.77)

From the table we have, Mean=14.91& Std. deviation = 5.291

The value of t at 95 % level of confidence & 63df is 1.998

We know the confidence interval,

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CI = ± t =14.91 ± (1.998)*( 5.291/7.94) = 14.91 ± 1.33 = (-86.42, -83.77)

Interpretation: Probability (-86.42≤ μ ≤ -83.77) = 0.95

The mean distance of the homes from the center of the city ranges from -86.42 miles to -83.77

miles we are 95 % confident about it.

In other words, the interval (from -86.42 miles, -83.77miles) contains the population mean with

probability 0.95. If we take a sample 100 times, this interval will contain the population 95 times.

III. 95 percent confidence interval for the proportion of homes with an attached garage

Garage Attached

Frequency Percent Valid Percent

Cumulative

Percent

Valid No 23 35.9 35.9 35.9

Yes 41 64.1 64.1 100.0

Total 64 100.0 100.0

We know, the sample proportion of homes with an attached garage,

= = 41/64 = 0.64

Thus we estimate 64 percent of the homes with an attached garage.

We determine the 95 % Confidence Interval for the proportion of homes with an attached garage

as

± z =.64±1.96√.64(1-.64)/64=.64 ± 0.1176= .5224, .7576

Interpretation: The proportion of homes with an attached garage ranges from (52-76) and we

are 95% confident about it.

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4. Refer to the Real Estate data, which report information on the homes sold in Denver,

Colorado, last year. [Chapter-10]

I. A recent article in the Denver Post indicated that the mean selling price of the homes in the

area is more than $2200. Can we conclude that the mean selling price in the Denver area is more

than $2200? Use the .01 significance level. What is the p-value?

II. The same article reported the mean size was more than 2100 square feet. Can we conclude

that the mean size of homes sold in the Denver area is more than 2100 square feet? Use the .01

significance level. What is the p-value?

Answer to the question No.4

i. Hypothesis testing:

Step1: State the Null Hypothesis (H0) and Alternative Hypothesis (H1)

H0 : μ ≤ $ 2200 i.e.; mean selling price of homes in Denver is not more than $2200

H1 : μ > $ 2200 i.e.; mean selling price of homes in Denver is more than $2200

Step 2: select the level of significance

Here, the level of significance is, α = .01

Step 3: Determine the appropriate test statistic

t-test statistic will be used here.

Step 4: Formulate the decision rule

If p-value < α – value (or calculated value is greater than critical value), H0 is rejected

If p-value > α- value (or calculated value is less than critical value), H0 is accepted

Step 5: Select the sample, perform the calculations and make a decision

From SPSS output, we get, One-Sample Statistics


Std. Error

Mean

Selling Price in TK.

000 (Thousand)64 2192.659 462.2481 57.7810

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One-Sample Test

Test Value = 2200


Mean

Difference


of the Difference

Lower Upper

Selling Price in TK.

000 (Thousand)-.127 63 .899 -7.3406 -160.815 146.134

We get, p-value=.899/2=.449

Decision: Since the computed value of the test statistic (t= -.127) is less than the critical value

(2.387) and P-value (.449) > α (.01). So we accept H0.

Comment:

So, we can conclude that the mean selling price of homes in Denver is not more than 2200

ii. Hypothesis testing:

Step1: State the Null Hypothesis (H0) and Alternative Hypothesis (H1)

H0: µ 2100

H1: µ> 2100

Here, H0 = mean size of the homes sold in the Denver area is not more than 2100 square feet

Here, H1 = mean size of the homes sold in the Denver area is more than 2100 square feet

Step 2: select the level of significance

Here the level of significance is, α = .01

Step 3: Determine the appropriate test statistic

t-test statistic will be used here.

Step 4: Formulate the decision rule

If p-value < α- value, H0 will be rejected

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If p-value > α- value, H0 will be accepted.

Step 5: Select the sample, perform the calculations and make a decision

From SPSS output, One-Sample Statistics


Std. Error

Mean

Size of the Home

in Square Feet64 2223.44 264.120 33.015

One-Sample Test

Test Value = 2100


Mean

Difference


of the Difference

Lower Upper

Size of the Home

in Square Feet3.739 63 .000 123.438 35.74 211.13

So, p-value = .001/2=.0005

Decision: Since P-value (0.0005) < α (.01) and (Calculated value, 3.739is larger than the critical

value or tabulated value 2.387). So we reject H0.

Comment:So, we can conclude that mean size of the homes sold in the Denver area is more than 2100

square feet.

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Statistic Lab