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 suburbs. A subset of this data, relating to rental apartments in these suburbs is included below. The variables 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).

The values are;

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 3)

The students were interested in the proportion of rental apartments in these suburbs that were leased as furnished apartments, and whether this varied with the number of bedrooms in these apartments. To investigate further whether the proportions of furnished apartments differ between apartments with different numbers of bedrooms, it is useful to test formally whether the number of bedrooms in an apartment and whether it is furnished or not are independent.

a) Test whether the number of bedrooms in an apartment and whether it is furnished or not is independent.

b) State the null hypothesis, the relevant form of the test statistic and the approximate distribution of the test statistic for carrying out this text.

c) Perform a hypothesis test with using α = 0.05, of whether the proportions of furnished apartments vary across number of bedrooms, that is, whether the furnishing status of an apartment is independent of the number of bedrooms in the apartment.

Include the Following:

i) The table of expected frequencies

ii) The observed value of the test statistic

iii) The relevant degrees of freedom for the distribution of the test statistic

IV) The resulting p-value for the test, or a rejection region

Conclude the test by interpreting the p-value (or rejection region and your observed test statistic) in terms of the original question discussed above

Pls do this question with R code !!!!

Answer 1

Part (a)

To Test whether the number of bedrooms in an apartment and whether it is furnished or not

are independent, as a first preparatory step, we construct the following (3 x 2) contingency table:

Number of Bedrooms	Furnished	Not Furnished	Total
1	5	11	16
2	23	48	63
3	4	20	24
Total	32	71	103

DONE

Part (b)

Hypotheses

Null H₀: The number of bedrooms in an apartment and whether it is furnished or not are independent, Vs

Alternative H₁: The two attributes are not independent

Test Statistic

χ² = ∑(i = 1 to r, j = 1 to c){(Oij - Eij)²/Eij}, where Oij and Eij are respectively, the observed and the expected frequencies of the ijth cell, r =number of rows and c = number of columns of the contingency table.

Distribution

Under H₀, χ² ~ χ²_{n = 2}, where n = {(r - 1)(s - 1)}

DONE

Part (c)

Calculations Summary

χ2cal =	3.195048	p-value =	0.202397
χ2crit =	5.991465	for DF =	2	α =	0.05

Since χ²_cal < χ²_crit, H₀ is accepted. [Equivalently, H₀ is accepted since p-value > α ]

Conclusion

There is sufficient evidence to suggest that whether an apartment is furnished or not does not depend on the number of bedrooms in the apartment.

DONE

Calculations Details

Oij (given)
	Oi1	O12	Oi.
O1i	5	11	16
O2j	23	40	63
O3j	4	20	24
O.i	32	71	103
Eij
	Ei1	Ei2	Ei.
E1j	4.970874	11.02912621	16
E2j	19.57282	43.42718447	63
E3j	7.456311	16.54368932	24
E.i	32	71	103
chi-square
	1	2	Total
1	0.000171	7.69178E-05	0.000248
2	0.600097	0.27046638	0.870564
3	1.602144	0.722093076	2.324237
Total	2.202412	0.992636374	3.195048