Question

In 2014, a group of students was interested in investigating prices of rental accommodation in suburbs of Brisbane that are close to the CBD, and collected information on a total of 200 randomly chosen dwellings in four inner western surbubs.

A subset of this data, relating to rental apartments in these suburbs, is provided. The variables in this file are:

• per week: weekly rental price for the apartment ($);

• Bedrooms: number of bedrooms in the apartment;

• Sqm: size of the apartment (m²)

• Furnished: whether the apartment was furnished or not (Yes/No).

Data: Per: week,Bedrooms,Furnished

265,2,59,No

305,2,70,No

300,1,72,No

320,3,66,No

340,2,113,Yes

330,2,58,Yes

355,2,63,No

345,2,57,Yes

355,2,61,No

360,2,114,Yes

355,2,75,Yes

360,2,68,No

365,2,64,No

370,1,69,No

390,2,73,Yes

380,2,85,Yes

390,2,56,Yes

370,2,56,Yes

385,2,59,Yes

380,2,65,Yes

385,2,62,Yes

400,2,65,No

415,2,69,Yes

400,3,63,No

405,3,70,No

420,2,77,No

435,2,84,Yes

435,2,83,Yes

455,2,73,Yes

450,2,72,Yes

485,2,68,No

500,2,76,Yes

535,2,97,No

290,1,60,No

305,1,63,Yes

330,2,65,No

310,2,70,No

335,2,64,No

330,2,62,No

345,2,79,No

355,1,81,No

340,2,66,No

345,1,60,No

345,2,64,No

355,2,73,No

385,2,61,No

380,2,78,No

405,2,81,No

410,2,76,Yes

430,2,80,No

440,2,61,No

450,3,86,No

485,3,91,No

500,1,87,No

545,1,97,Yes

345,3,86,No

400,2,72,No

400,2,74,No

480,2,73,Yes

755,3,87,No

760,3,77,No

770,3,113,No

824,2,109,No

860,3,104,No

295,1,70,No

290,1,54,No

295,1,61,No

325,1,61,No

340,2,56,No

355,2,61,No

365,2,95,No

420,1,75,No

420,2,66,No

440,2,74,No

480,3,72,No

465,3,87,No

470,1,87,Yes

490,1,81,Yes

495,2,76,No

505,2,97,No

530,2,77,No

545,2,97,No

560,2,79,No

550,2,78,No

560,3,75,No

565,1,96,Yes

580,2,85,Yes

605,3,84,No

605,2,93,Yes

610,2,78,Yes

620,2,87,No

665,2,88,No

700,2,80,No

750,3,97,Yes

740,3,124,No

805,3,101,No

860,3,98,No

960,3,123,Yes

990,3,102,Yes

1195,3,133,No

1190,3,137,No

1405,3,148,Yes

1490,3,154,No

Question 1.

One of the students was particularly interested in three-bedroom apartments and believed that the average rental price for a three-bedroom apartments in this area in 2014, when the data were collected was $700 per week. By hand (you may use R to obtain any relevant statistics you need from the data), carry out a formal statistical test (as per the following steps) to assess whether the data in the sample provide any evidence to contradict this assumption.

(a) State the null hypothesis and the relevant form of the test statistic (and its distribution) for this test.

(b) Obtain the relevant sample statistics to carry out this test.
(c) Calculate the observed value of the test statistic and use it to find the p-value for this test.

(d) Draw a conclusion and interpret it in the context of the statement “the average rent for a three-bedroom apartments in this area was $700 per week in 2014”.

Answer 1

Since the student's claim is that the average rental price for a three-bedroom apartments in this area in 2014, when the data were collected was $700 per week. ​We use one sample t-test.

Because One Sample t Test determines whether the sample mean is statistically different from a known or hypothesized population mean. To perform this statistical test we use only weekly rental price for the apartment ($) for 3 bed room apartments. Thus the data to be used for one sample t-test with total sample size (n) = 24 is as follows:

320,400,405,450,485,345,755,760,770,860,480,465,560,605,750,740,805,860,960,990,1195,1190,

1405,1490

(a) HYPOTHESIS:

​: The average rental price for a three-bedroom apartments in this area in 2014 is equal to 700$ per week.

​ (in dollars)

​: The average rental price for a three-bedroom apartments in this area in 2014 is not equal to 700$ per week.

(in dollars)

LEVEL OF SIGNIFICANCE:

   Let us consider 5% level of significance. That is,

TEST STATISTIC:

   The test statistic for a One Sample t Test is denoted t, which is calculated using the following formula:

which follows t distribution with (n-1) degrees of freedom and 5% level of significance.

Here ​ (proposed constant for the population mean)

is sample mean

n = 24 (sample size)

s is sample standard deviation.

If the calculated t value > critical t value, then we reject the null hypothesis.

(b) RELEVANT SAMPLE STATISTIC:

MEAN OF SAMPLE :

   Mean = (320+400+405+450+485+345+755+760+770+860+480+465+560+605+

   750+740+805+860+960+990+1195+1190+1405+1490)/24

= 751.88

STANDARD DEVIATION (s):

Mean = 751.88

320	-431.88	186520.3
400	-351.88	123819.5
405	-346.88	120325.7
450	-301.88	91131.5
485	-266.88	71224.9
345	-406.88	165551.3
755	3.12	9.7344
760	8.12	65.93
770	18.12	328.33
860	108.12	11689.9
480	-271.88	73918.7
465	-286.88	82300.1
560	-191.88	36817.9
605	-146.88	21573.7
750	-1.88	3.534
740	-11.88	141.13
805	53.12	2821.7
860	108.12	11689.9
960	208.12	43313.9
990	238.12	56701.13
1195	443.12	196355.3
1190	438.12	191949.1
1405	653.12	426565.7
1490	738.12	544821.1

Standard deviation (s) =

   =

   =

   =

(c) OBSERVED TEST STATISTIC:

  

  

  

P-VALUE:

To find the p-value using t-table. We look into the row corresponding to 23 degrees of freedom and look for the upper and lower range where calculated t-statistic value lie. Thus t statistic value lie between statistic 0.685 and 0.858. Look into corresponding p-value (column value). Thus the p value for the calculated t-statistic lie between 0.40 and 0.50 according to t table. The exact p-value is 0.437 (using R).

R code and Output:

(d) CONCLUSION AND INTERPRETATION:

t TABLE:

TABLE CRITICAL VALUE:

The t table value with (n-1=24-1=23) degrees of freedom at 5% level of significance is 2.069. Since the calculated t statistic (0.78) is less than the critical table value 2.069, we fail to reject H_o . Thus we conclude that the average rent for a three-bedroom apartments in this area is not $700 per week in 2014.

And since the p value for the calculated t-statistic lie between 0.40 and 0.50 according to t table. And the exact p-value is 0.437 (using R). Thus p-value is greater than 0.05, we fail to reject H_o and conclude that the average rent for a three-bedroom apartments in this area is not $700 per week in 2014.