Question

Use the Housing_Interest Rate database

YEAR	MONTH	FIXED_RATE%	STARTS in $100	# Houses SOLD
1990	1	9.81	1551	45
1990	2	9.97	1437	50
1990	3	10.03	1289	58
1990	4	10.14	1248	52
1990	5	10.22	1212	50
1990	6	10.21	1177	50
1990	7	10.2	1171	46
1990	8	9.99	1115	46
1990	9	9.99	1110	38
1990	10	10.06	1014	37
1990	11	10.11	1145	34
1990	12	9.87	969	29
1991	1	9.75	798	30
1991	2	9.62	965	40
1991	3	9.45	921	51
1991	4	9.47	1001	50
1991	5	9.52	996	47
1991	6	9.49	1036	47
1991	7	9.49	1063	43
1991	8	9.52	1049	46
1991	9	9.33	1015	37
1991	10	9.1	1079	41
1991	11	8.77	1103	39
1991	12	8.58	1079	36
1992	1	8.35	1176	48
1992	2	8.46	1250	55
1992	3	8.65	1297	56
1992	4	8.71	1099	53
1992	5	8.68	1214	52
1992	6	8.52	1145	53
1992	7	8.28	1139	52
1992	8	8.09	1226	56
1992	9	7.92	1186	51
1992	10	7.92	1244	48
1992	11	8.06	1214	42
1992	12	8.18	1227	42
1993	1	8.08	1210	44
1993	2	7.86	1210	50
1993	3	7.67	1083	60
1993	4	7.56	1258	66
1993	5	7.48	1260	58
1993	6	7.48	1280	59
1993	7	7.34	1254	55
1993	8	7.24	1300	57
1993	9	7.08	1343	57
1993	10	6.93	1392	56
1993	11	6.99	1376	53
1993	12	7.2	1533	51
1994	1	7.19	1272	46
1994	2	7.14	1337	58
1994	3	7.32	1564	74
1994	4	7.68	1465	65
1994	5	8.15	1526	65
1994	6	8.33	1409	55
1994	7	8.36	1439	52
1994	8	8.5	1450	59
1994	9	8.5	1474	54
1994	10	8.64	1450	57
1994	11	8.79	1511	45
1994	12	8.9	1455	40
1995	1	9.06	1407	47
1995	2	8.96	1316	47
1995	3	8.82	1249	60
1995	4	8.6	1267	58
1995	5	8.3	1314	63
1995	6	7.88	1281	64
1995	7	7.76	1461	64
1995	8	7.88	1416	63
1995	9	7.82	1369	54
1995	10	7.71	1369	54
1995	11	7.63	1452	46
1995	12	7.51	1431	45
1996	1	7.28	1467	54
1996	2	7.24	1491	68
1996	3	7.47	1424	70
1996	4	7.82	1516	70
1996	5	8.05	1504	69
1996	6	8.17	1467	65
1996	7	8.27	1472	66
1996	8	8.19	1557	73
1996	9	8.2	1475	62
1996	10	8.12	1392	56
1996	11	7.92	1489	54
1996	12	7.77	1370	51
1997	1	7.87	1355	61
1997	2	7.87	1486	69
1997	3	7.91	1457	81
1997	4	8.1	1492	70
1997	5	8.14	1442	71
1997	6	8	1494	71
1997	7	7.79	1437	69
1997	8	7.69	1390	72
1997	9	7.69	1546	67
1997	10	7.57	1520	62
1997	11	7.5	1510	61
1997	12	7.41	1566	51
1998	1	7.24	1525	64
1998	2	7.19	1584	75
1998	3	7.19	1567	81
1998	4	7.21	1540	82
1998	5	7.21	1536	82
1998	6	7.2	1641	83
1998	7	7.13	1698	75
1998	8	7.09	1614	75
1998	9	6.97	1582	68
1998	10	6.82	1715	69
1998	11	6.85	1660	70
1998	12	6.88	1792	61
1999	1	6.89	1748	67
1999	2	6.92	1670	76
1999	3	7.01	1710	84
1999	4	7.05	1553	86
1999	5	7.09	1611	80
1999	6	7.34	1559	82
1999	7	7.59	1669	78
1999	8	7.79	1648	78
1999	9	7.87	1635	65
1999	10	7.87	1608	67
1999	11	7.87	1648	61
1999	12	7.9	1708	57
2000	1	8.08	1636	67
2000	2	8.27	1737	78
2000	3	8.31	1604	88
2000	4	8.27	1626	78
2000	5	8.35	1575	77
2000	6	8.43	1559	71
2000	7	8.29	1463	76
2000	8	8.16	1541	73
2000	9	8.03	1507	70
2000	10	7.95	1549	71
2000	11	7.85	1551	63
2000	12	7.68	1532	65
2001	1	7.31	1600	72
2001	2	7.13	1625	85
2001	3	7.06	1590	94
2001	4	7.09	1649	84
2001	5	7.18	1605	80
2001	6	7.21	1636	79
2001	7	7.21	1670	76
2001	8	7.13	1567	74
2001	9	6.97	1562	66
2001	10	6.76	1540	66
2001	11	6.67	1602	67
2001	12	6.89	1568	66
2002	1	7.02	1698	66
2002	2	6.98	1829	84
2002	3	6.98	1642	90
2002	4	7.11	1592	86
2002	5	6.99	1764	88
2002	6	6.87	1717	84
2002	7	6.72	1655	82
2002	8	6.53	1633	90
2002	9	6.36	1804	82
2002	10	6.23	1648	77
2002	11	6.2	1753	73
2002	12	6.21	1788	70
2003	1	6.09	1853	76
2003	2	6.02	1629	82
2003	3	5.9	1726	98
2003	4	5.9	1643	91
2003	5	5.74	1751	101
2003	6	5.5	1867	107
2003	7	5.53	1897	99
2003	8	5.88	1833	105
2003	9	6.19	1939	90
2003	10	6.05	1967	88
2003	11	6.06	2083	76
2003	12	6	2057	75
2004	1	5.92	1927	89
2004	2	5.85	1852	102
2004	3	5.71	2007	123
2004	4	5.72	1968	109
2004	5	6.07	1974	115
2004	6	6.25	1827	105
2004	7	6.26	1986	96
2004	8	6.1	2025	102
2004	9	5.9	1912	94
2004	10	5.91	2062	101
2004	11	5.89	1807	84
2004	12	5.9	2050	83
2005	1	5.9	2188	92
2005	2	5.9	2228	109
2005	3	5.98	1836	128
2005	4	6.09	2038	122

The objective is to compare FIXED_RATE between two periods; before 2000 and afteryear 2000 including 2000

i) Use a random generating procedure to draw a random sample of size 80 with respect to the “before and after year 2000 “factor.

• Using your sample data:
• Do Boxplot and the descriptive statistics of FIXED_RATE for before and after 2000.

Answer 1

Note- we will use R -software to draw a random sample of size 80 with respect to the “before and after year 2000 “factor. Any other software of statistical book can be used for the same purpose .

We will import data into R

Now we need to compare FIXED_RATE between two periods; before 2000 and afteryear 2000 including 2000 , so we will take only first 3 columns of given data - ( i.e "YEAR" , "MONTH" , " FIXED_RATE% " )

Now we need a random generating procedure to draw a random sample of size 80 with respect to the “before and after year 2000 “factor.

Number of observation before year 2000 - 120

Number of observation after year 2000 including 2000 - 64

Here , in population we have more observation for before year 2000 ( = 120 ) and less observation for after year 2000 ( = 64 ) . hence population with respect to the two grops “beforeyear 2000" and "after year 2000 “ factor is not in proportion , so takinjg random samples from whole population may give more samples from one group and less samples from another group . So we need to take 80 random sample in such a way that proportion of sample from two group is same to that of population proportion .

Ration of Observation before 2000 and after 2000 is = 120 / 64 = 1.875

Thus for a sample of size 80 ration of before 2000 and after 2000 is = 52 /28 = 1.857143

i.e in a sample of size 80 we must include 52 random samples of observation before 2000 and 28 random samples of observation after 2000 such that total random samples would be 52 + 28 = 80

So we take 52 random sample of before year 2000 and 28 random sample of after year 2000 including year 2000

Such that our total samples size would be 80 are Proportion of Population size would not get voilated

R- code and output

year=scan("clipboard")         # to import year
month=scan("clipboard")      # to import months
F_R=scan("clipboard")         # to import Fixed Rate

before=(which(year<2000))         # to find observation before year 2000
after=(which(year>=2000))         # to find observation after year 2000

spl_52_before=sample(before,52) # sample size of 52 from before year 2000
spl_28_after=sample(after,28)           # sample size of 28from after year 2000

SAMPLE=c(spl_52_before,spl_28_after)                      # combining 80 sample numbers
spl_YEAR=c(year[spl_52_before],year[spl_28_after])    # coorespong years to sample numbers

   

F_R_before=F_R[spl_52_before]     # coorespong FIXED RATE to sample numbers before 2000
F_R_after=F_R[spl_28_after]          # coorespong FIXED RATE to sample numbers after 2000
F_R_spl=c(F_R_before,F_R_after)   # combining FIXED RATE corresponding samples

# So these are sample numbers and corresponding year

kable(data.frame(SR_No.=1:80,"\t",SAMPLE,"\t","YEAR"=spl_YEAR,"\t","FIXED RATE"=F_R_spl))

|SRNo| |SAMPLE|    |YEAR| |FIXED.RATE|

|      1|       |     70|        | 1995|        |       7.71 |
|      2|       |     93|        | 1997|        |       7.69 |
|      3|       |      1|        | 1990|         |      9.81 |
|      4|       |     69|        | 1995|        |       7.82 |
|      5|     |     57|        | 1994|        |       8.50|
|      6|     |     91|        | 1997|        |       7.79|
|      7|     |      2 |        | 1990|        |       9.97|
|      8|     |     17|        | 1991|        |       9.52|
|      9|     |     90|        | 1997|        |       8.00|
|     10|      |     61|        | 1995|        |       9.06|
|     11|      |     54|        | 1994|        |       8.33|
|     12|      |     78|        | 1996|        |       8.17|
|     13|      |     58|        | 1994|        |       8.64|
|     14|      |      7 |        | 1990|        |      10.20|
|     15|      |     19|        | 1991|        |       9.49|
|     16|      |     62|        | 1995|        |       8.96|
|     17|      |     20|        | 1991|        |       9.52|
|     18|      |     10|        | 1990|        |      10.06|
|     19|      |    119|        | 1999|        |       7.87|
|     20|      |     99|        | 1998|        |       7.19|
|     21|      |     26|        | 1992|        |       8.46|
|     22|      |     32|        | 1992|        |       8.09|
|     23|      |   102|        | 1998|        |       7.20|
|     24|      |     73|        | 1996|        |       7.28|
|     25|      |     44|        | 1993|        |       7.24|
|     26|      |     42|        | 1993|        |       7.48|
|     27|      |     23|        | 1991|        |       8.77|
|     28|      |     63|        | 1995|        |       8.82|
|     29|      |      9 |        | 1990|        |       9.99|
|     30|      |    46|       | 1993|        |       6.93|
|     31|      |   118|       | 1999|        |       7.87|
|     32|      |      8 |        | 1990|        |       9.99|
|     33|      |     95|        | 1997|        |       7.50|
|     34|      |     18|        | 1991|        |       9.49|
|     35|      |     45|        | 1993|        |       7.08|
|     36|      |    117|        | 1999|        |       7.87|
|     37|      |     27|        | 1992|        |       8.65|
|     38|      |     48|        | 1993|        |       7.20|
|     39|      |     38|        | 1993|        |       7.86|
|     40|      |     37|        | 1993|        |       8.08|
|     41|      |     50|        | 1994|        |       7.14|
|     42|      |     43|        | 1993|        |       7.34|
|     43|      |     74|        | 1996|        |       7.24|
|     44|      |    104|        | 1998|        |       7.09|
|     45|      |     71|        | 1995|        |       7.63|
|     46|      |     28|        | 1992|        |       8.71|
|     47|      |     53|        | 1994|        |       8.15|
|     48|      |     75|        | 1996|        |       7.47|
|     49|      |     12|        | 1990|        |       9.87|
|     50|      |     81|        | 1996|        |       8.20|
|     51|      |    111|        | 1999|        |       7.01|
|     52|      |     56|        | 1994|        |       8.50|
|     53|      |    125|        | 2000|        |       8.35|
|     54|      |    124|        | 2000|        |       8.27|
|     55|      |    176|        | 2004|        |       6.10|
|     56|      |    122|        | 2000|        |       8.27|
|     57|      |    143|        | 2001|        |       6.67|
|     58|      |    153|        | 2002|        |       6.36|
|     59|      |    170|        | 2004|        |       5.85|
|     60|      |    146|        | 2002|        |       6.98|
|     61|      |    173|        | 2004|        |       6.07|
|     62|      |    147|        | 2002|        |       6.98|
|     63|      |    155|        | 2002|        |       6.20|
|     64|      |    178|        | 2004|        |       5.91|
|     65|      |    166|        | 2003|        |       6.05|
|     66|      |    134|        | 2001|        |       7.13|
|     67|      |    181|        | 2005|        |       5.90|
|     68|      |    136|        | 2001|        |       7.09|
|     69|      |    133|        | 2001|        |       7.31|
|     70|      |    171|        | 2004|        |       5.71|
|     71|      |    130|        | 2000|        |       7.95|
|     72|      |    128|        | 2000|        |       8.16|
|     73|      |    163|        | 2003|        |       5.53|
|     74|      |    174|        | 2004|        |       6.25|
|     75|      |    157|        | 2003|        |       6.09|
|     76|      |    137|        | 2001|        |       7.18|
|     77|      |    129|        | 2000|        |       8.03|
|     78|      |    150|        | 2002|        |       6.87|
|     79|      |    135|        | 2001|        |       7.06|
|     80|      |    151|        | 2002|        |       6.72|

Now we have seperate samples of size 80 for two periods; before 2000 and after year 2000 including 2000
Now we need to draw boxplot for corresponding FIXED RATE%

R- code and output

>summary(F_R_before)      # summary of samples of Fixed Rate% before year 2000
   Min.     1st Qu.    Median   Mean 3rd Qu.     Max.
6.930    7.478    8.085 8.279    8.855     10.200

> summary(F_R_after)        # summary of samples of Fixed Rate% after year 2000
   Min.     1st Qu. Median     Mean    3rd Qu.    Max.
5.530     6.085   6.795    6.823     7.212     8.350

# box plot for samples of FIXED RATE % before year 2000

>boxplot(F_R_before,main="Before 2000",col=2)

# box plot for samples of FIXED RATE % after year 2000

>boxplot(F_R_after,main="After 2000",col=3)

# and Combine Boxplot of fixed rate begore year 2000 and after year 2000 is

>boxplot(F_R_before,F_R_after,names=c("Before 2000","After 2000") ,
+          main="Box-Plot",col=c(2,3))

Hence from box plot we can see that Fixed Rate % have decreased after year 2000

So , comparing FIXED_RATE between two periods; before 2000 and after year 2000 including 2000 we conclude that FIXED_RATE of second period i.e after year 2000 including 2000 has decreased significantly .