Question

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?

Calculate the Chi Square ..the critical value ..the p -value and the margin error!

REGION ID

1= Crete
2=Southern Aegean Islands
3=Ionian Islands

STARS	Region_ID
1	1
1	1
1	1
1	1
1	1
1	1
1	1
1	1
1	1
1	2
1	2
1	2
1	2
1	2
1	2
1	2
1	2
1	2
1	2
1	3
1	3
1	3
1	3
1	3
1	3
1	3
2	1
2	1
2	1
2	1
2	1
2	1
2	1
2	1
2	1
2	2
2	2
2	2
2	2
2	2
2	2
2	2
2	2
2	2
2	3
2	3
2	3
2	3
2	3
2	3
2	3
2	3
2	3
2	3
3	1
3	1
3	1
3	1
3	1
3	1
3	1
3	1
3	1
3	1
3	2
3	2
3	2
3	2
3	2
3	2
3	2
3	2
3	2
3	2
3	3
3	3
3	3
3	3
3	3
3	3
3	3
3	3
3	3
3	3
4	1
4	1
4	1
4	1
4	1
4	1
4	1
4	1
4	1
4	1
4	2
4	2
4	2
4	2
4	2
4	2
4	2
4	2
4	2
4	2
4	3
4	3
4	3
4	3
4	3
4	3
4	3
4	3
5	1
5	1
5	1
5	1
5	1
5	1
5	1
5	2
5	2
5	2
5	2
5	2
5	2
5	3
5	3
5	3
5	3
5	3
5	3
5	3
5	3
5	3
5	3

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?

Answer 1

The contingency table is:

5 Star	Crete	Elsewhere	Total
Yes	7	16	23
No	38	74	112
Total	45	90	135

3.1 State the null and the alternative hypotheses.

The hypothesis being tested is:

H₀: The percentage composition of 5 Stars hotels is the same in Crete and elsewhere (Southern Aegean Islands and Ionian Islands).

H_a: The percentage composition of 5 Stars hotels is not the same in Crete and elsewhere (Southern Aegean Islands and Ionian Islands).
3.2 Test the null hypothesis at α = 5%.

The output is:

		Crete	Elsewhere	Total
Yes	Observed	7	16	23
	Expected	7.67	15.33	23.00
	O - E	-0.67	0.67	0.00
	(O - E)² / E	0.06	0.03	0.09
No	Observed	38	74	112
	Expected	37.33	74.67	112.00
	O - E	0.67	-0.67	0.00
	(O - E)² / E	0.01	0.01	0.02
Total	Observed	45	90	135
	Expected	45.00	90.00	135.00
	O - E	0.00	0.00	0.00
	(O - E)² / E	0.07	0.03	0.10
		.10	chi-square
		1	df
		.7461	p-value

Since the p-value (0.7461) is greater than the significance level (0.05), we cannot reject the null hypothesis.

3.3 What is your conclusion?

Therefore, we cannot conclude that the percentage composition of 5 Stars hotels is not the same in Crete and elsewhere (Southern Aegean Islands and 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?

p1	p2	pc
0.1556	0.1778	0.1704	p (as decimal)
7/45	16/90	23/135	p (as fraction)
7.	16.	23.	X
45	90	135	n
	-0.0222	difference
	0.0686	std. error
	-0.1543	confidence interval 95.% lower
	0.1099	confidence interval 95.% upper
	0.1321	margin of error

The 95% confidence interval estimate of the difference (π1 - π2) is between -0.1543 and 0.1099.

Since the confidence interval contains 0, we cannot conclude that there is a difference between the population proportion of 5 stars hotels in Crete and the population proportion of 5 stars hotels elsewhere.