Question

In: Statistics and Probability

Subject 3 George who is the General Director of the Research Institute for Tourism in Greece,...

Subject 3

George who is the General Director of the Research Institute for Tourism in Greece, believes that the percentage composition of 5 Stars hotels is not the same in Crete and elsewhere (Southern Aegean Islands and Ionian Islands). In order to test this statement, create two additional binary variables as follows:

5STARS: Yes (Code 1) when STARS = 5; No (Code 0) when STARS ≠ 5.

CRETE: Yes (Code 1) when Region_ID = 1; No (Code 0) when Region_ID ≠ 1.

3.1 State the null and the alternative hypotheses.

3.2 Test the null hypothesis at α = 5%.

3.3 What is your conclusion?

3.4 Let π1 denote the population proportion of 5 stars hotels in Crete and π2 the population proportion of 5 stars hotels elsewhere. Set up a symmetric 95% confidence interval estimate of the difference (π1 - π2). What is your main conclusion?

STARS	Total_Rooms	Region_ID	ARR_MAY	ARR_AUG	L_COST
5	412	1	95	160	2.165.000
5	313	1	94	173	2.214.985
5	265	1	81	174	1.393.550
5	204	1	131	225	2.460.634
5	172	1	90	195	1.151.600
5	133	1	71	136	801.469
5	127	1	85	114	1.072.000
4	322	1	70	159	1.608.013
4	241	1	64	109	793.009
4	172	1	68	148	1.383.854
4	121	1	64	132	494.566
4	70	1	59	128	437.684
4	65	1	25	63	83.000
3	93	1	76	130	626.000
3	75	1	40	60	37.735
3	69	1	60	70	256.658
3	66	1	51	65	230.000
3	54	1	65	90	200.000
2	68	1	45	55	199.000
1	57	1	35	90	11.720
4	38	1	22	51	59.200
4	27	1	70	100	130.000
3	47	1	60	120	255.020
3	32	1	40	60	3.500
3	27	1	48	55	20.906
2	48	1	52	60	284.569
2	39	1	53	104	107.447
2	35	1	80	110	64.702
2	23	1	40	50	6.500
1	25	1	59	128	156.316
4	10	1	90	105	15.950
3	18	1	94	104	722.069
2	17	1	29	53	6.121
2	29	1	26	44	30.000
1	21	1	42	54	5.700
1	23	1	30	35	50.237
2	15	1	47	50	19.670
1	8	1	31	49	7.888
1	20	1	35	45
1	11	1	40	55
1	15	1	40	55	3.500
1	18	1	35	40	112.181
3	23	1	40	55
4	10	1	57	97	30.000
2	26	1	35	40	3.575
5	306	2	113	235	2.074.000
5	240	2	61	132	1.312.601
5	330	2	112	240	434.237
5	139	2	100	130	495.000
4	353	2	87	152	1.511.457
4	324	2	112	211	1.800.000
4	276	2	95	160	2.050.000
4	221	2	47	102	623.117
4	200	2	77	178	796.026
4	117	2	48	91	360.000
3	170	2	60	104	538.848
3	122	2	25	33	568.536
5	57	2	68	140	300.000
4	62	2	55	75	249.205
3	98	2	38	75	150.000
3	75	2	45	70	220.000
3	62	2	45	90	50.302
5	50	2	100	180	517.729
4	27	2	180	250	51.000
3	44	2	38	84	75.704
3	33	2	99	218	271.724
3	25	2	45	95	118.049
2	42	2	28	40
2	30	2	30	55	40.000
1	44	2	16	35
3	10	2	40	70	10.000
2	18	2	60	100	10.000
1	18	2	16	20
2	73	2	22	41	70.000
2	21	2	55	100	12.000
1	22	2	40	100	20.000
1	25	2	80	120	36.277
1	25	2	80	120	36.277
1	31	2	18	35	10.450
3	16	2	80	100	14.300
2	15	2	30	45	4.296
1	12	2	40	65
1	11	2	30	50
1	16	2	25	70	379.498
1	22	2	30	35	1.520
4	12	2	215	265	45.000
4	34	2	133	218	96.619
2	37	2	35	95	270.000
2	25	2	100	150	60.000
2	10	2	70	100	12.500
5	270	3	60	90	1.934.820
5	261	3	119	211	3.000.000
5	219	3	93	162	1.675.995
5	280	3	81	138	903.000
5	378	3	44	128	2.429.367
5	181	3	100	187	1.143.850
5	166	3	98	183	900.000
5	119	3	100	150	600.000
5	174	3	102	211	2.500.000
5	124	3	103	160	1.103.939
4	112	3	40	56	363.825
4	227	3	69	123	1.538.000
4	161	3	112	213	1.370.968
4	216	3	80	124	1.339.903
3	102	3	53	91	173.481
4	96	3	73	134	210.000
4	97	3	94	120	441.737
4	56	3	70	100	96.000
3	72	3	40	75	177.833
3	62	3	50	90	252.390
3	78	3	70	120	377.182
3	74	3	80	95	111.000
3	33	3	85	120	238.000
3	30	3	50	80	45.000
3	39	3	30	68	50.000
3	32	3	30	100	40.000
2	25	3	32	55	61.766
2	41	3	50	90	166.903
2	24	3	70	120	116.056
2	49	3	30	73	41.000
2	43	3	94	120	195.821
4	9	3	100	180
2	20	3	70	120	96.713
2	32	3	19	45	6.500
2	14	3	35	70	5.500
2	14	3	50	80	4.000
1	13	3	25	45	15.000
1	13	3	30	50	9.500
2	53	3	55	80	48.200
3	11	3	95	120	3.000
1	16	3	25	31	27.084
1	21	3	16	40	30.000
1	21	3	16	40	20.000
1	46	3	19	23	43.549
1	21	3	30	40	10.000

Expert Solution

3.1 State the null and the alternative hypotheses.

Let π₁ be the percentage composition of 5 Stars hotels in Crete

Let π₂ be the percentage composition of 5 Stars hotels in Southern Aegean Islands and the Ionian Islands

The hypothesis being tested is:

H₀: π₁ = π₂

H₁: π₁ ≠ π₂

3.2 Test the null hypothesis at α = 5%.

The output is:

p1	p2	p_c
0.3043	0.6957	0.5	p (as decimal)
7/23	16/23	23/46	p (as fraction)
7.	16.	23.	X
23	23	46	n

	-0.391	difference
	0.	hypothesized difference
	0.1474	std. error
	-2.65	z
	.0080	p-value (two-tailed)

	-0.657	confidence interval 95.% lower
	-0.125	confidence interval 95.% upper
	0.2659	margin of error

Since the p-value (0.0080) is less than the significance level (0.05), we can reject the null hypothesis.

3.3 What is your conclusion?

Therefore, we can conclude that the percentage composition of 5 Stars hotels is not the same in Crete and elsewhere (Southern Aegean Islands and the Ionian Islands).

3.4 Let π1 denote the population proportion of 5 stars hotels in Crete and π2 the population proportion of 5 stars hotels elsewhere. Set up a symmetric 95% confidence interval estimate of the difference (π1 - π2). What is your main conclusion?

The symmetric 95% confidence interval estimate of the difference (π1 - π2) is between -0.657 and -0.125.

Since the confidence interval does not contain 0, we can conclude that the percentage composition of 5 Stars hotels is not the same in Crete and elsewhere (Southern Aegean Islands and the Ionian Islands).

orchestra answered 2 years ago

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