Use the aforementioned data set in order to:
Find an unbiased point estimate for the population mean of the average cost of the meal per person in Euros.
Set up a 95% confidence interval to estimate the population mean of the average cost of the meal per person in Euros.
On the basis only of your results in 1.2, do you think that the hypothesis that the mean of the average cost of the meal per person is 35 Euros should be rejected at α = 5% significance level? Explain.
Does the population of the average cost of the meal per person in Euros have to be normally distributed here? Explain.
Suppose that you use a 90% confidence interval estimate. How would your answers to 1.2 and 1.3 be affected?
Unbiased Point Estimate
The mean of the average cost of the meal per person is €36.74. The unbiased estimator of mean of the average cost of the meal per person is €37.1111.
Meal cost 
One Sample Ztest for estimating confidence interval 
Sum 
3674 
Mean (μ) 
36.74 
Unbiased estimator of mean (Xbar) 
37.11111111 
Standard deviation (s) 
9.203227365 
(Xbar  μ) 
0.371111111 
Count (n) 
100 
Squareroot of n 
10 
s/squareroot(n) 
0.920322737 
Confidence interval 
95% 
Level of significance 
5% 
Zstatistic 
1.959963985 
Confidence interval (95%) 
1.803799418 
Upper confidence limit 
38.91491053 
Lower confidence limit 
35.30731169 
The 95% confidence interval of average cost of meal per person is found to be (35.30731169, 38.91491053).
Meal cost 
One sample ttest (onetail) 
Xbar 
36.74 
Hypothesized mean (μ) 
35 
(Xbar  μ) 
1.74 
Standard deviation (s) 
9.203227365 
Count (n) 
100 
Degrees of freedom (d.f.) 
99 
Squareroot of n 
10 
s/squareroot(n) 
0.920322737 
tstatistics 
1.89064111 
pvalue 
0.030798262 
Null Hypothesis 
Meal cost per person is equal to €35 
Decision making 
Null Hypothesis rejected 
Null Hypothesis (H_{0}): Average Meal cost per person is equal to €35, that is, (μ = 35).
Alternative Hypothesis (H_{A}): Average Meal cost per person is unequal to €35, that is, (μ ≠ 35).
The tstatistic (1.89064111) is calculated as:
t = (sample mean – hypothesized mean)/ SE mean.
The calculated pvalue is 0.030798262 that is less than 0.05. Therefore, we reject null hypothesis of equality of average meal cost per person €35 at 5% level of significance.
The population of the average cost of the mean per person in Euros is not normally distributed. We use a normality test for determining the validity of assumption of normality. The pvalue less than 0.05 indicate that mean cost per person in Euros is not normally distributed.
Meal cost 
One sample Ztest for estimating confidence interval 
Sum 
3674 
Mean (μ) 
36.74 
Unbiased estimator of mean (Xbar) 
37.11111111 
Standard deviation (s) 
9.203227365 
Count (n) 
100 
Squareroot of n 
10 
s/squareroot(n) 
0.920322737 
Confidence interval 
90% 
Level of significance 
10% 
Zstatistic 
1.644853627 
Confidence interval (95%) 
1.513796191 
Upper confidence limit 
38.6249073 
Lower confidence limit 
35.59731492 
The 90% confidence interval of average cost of meal per person is found to be (35.59731492, 38.6249073).
Meal cost 
One sample ttest (onetail) 
Xbar 
36.74 
Hypothesized mean (μ) 
35 
(Xbar  μ) 
1.74 
Standard deviation (s) 
9.203227365 
Count (n) 
100 
Degrees of freedom (d.f.) 
99 
Squareroot of n 
10 
s/squareroot(n) 
0.920322737 
tstatistics 
1.89064111 
pvalue 
0.030798262 
Null Hypothesis 
Meal cost per person is equal to €35 
Decision making 
Null Hypothesis rejected 
The calculated pvalue is 0.030798262 that is less than 0.1. Therefore, we reject null hypothesis of equality of average meal cost per person €35 at 10% level of significance.
The minister of tourism in Greece insists that the mean of the average cost of the meal per person in Euros in Athens and Thessaloniki be less than €40 in order to attract more tourists.
The hypothesis are
Null hypothesis (H_{0}): The average meal cost per person is greater than or equals to €40, that is, (μ ≥ 40).
Alternative hypothesis (H_{A}): The average meal cost per person is less than €40, that is, (μ < 40).
The null hypothesis is to be tested at α = 5%.
Meal cost 
One sample ttest (onetail) 
Xbar 
36.74 
Hypothesized mean (μ) 
40 
(Xbar  μ) 
3.26 
Standard deviation (s) 
9.203227365 
Count (n) 
100 
Degrees of freedom (d.f.) 
99 
Squareroot of n 
10 
s/squareroot(n) 
0.920322737 
tstatistics 
3.542235643 
pvalue 
0.00030344 
Null Hypothesis 
Meal cost per person is less than €40 
Decision making 
Null Hypothesis rejected 
Based on outputs onesample ttest at 5% confidence interval, we can infer that null hypothesis of mean of meal cost per person greater than or equal to €40 could be rejected. Hence, we can conclude that minister’s requirement is satisfied here.
We create an additional binary variable by classifying the overall customer satisfaction as follows:
Cust_Sat_new: Overall Customer Satisfaction <=2.5; Satisfaction when Overall Customer Satisfaction >2.5.
Dissatisfied/Overall Customer 
Satisfied/Overall Customer (>2.5) 
2.33 
3.17 
2.33 
3.00 
2.33 
3.17 
2.33 
3.83 
2.50 
3.83 
2.00 
3.83 
2.17 
3.33 
1.83 
3.33 
2.00 
3.17 
1.83 
3.50 
2.00 
3.33 
2.50 
3.50 
2.33 
4.00 
3.33 

2.83 

3.50 

3.50 

3.33 

2.83 

3.50 

3.83 

2.83 

3.67 

3.17 

3.50 

3.17 

3.17 

3.50 

4.00 

3.17 

2.83 

3.17 

3.67 

3.67 

3.67 

3.33 

3.00 

3.00 

4.00 

3.50 

3.00 

3.33 

3.50 

3.17 

3.00 

3.33 

3.50 

3.17 

3.50 

3.00 

2.83 

3.17 

3.67 

3.00 

3.17 

3.00 

3.00 

3.33 

3.33 

3.00 

2.83 

3.83 

2.83 

3.17 

3.17 

3.50 

2.83 

2.83 

3.33 

2.67 

2.83 

2.67 

4.17 

2.83 

3.67 

3.00 

3.33 

3.00 

3.50 

3.17 

4.50 

3.00 

2.67 

3.33 

2.67 

4.00 

3.00 
Satisfied Customers 

Total (n) 
100 
Success proportion (p) 
0.87 
Failure proportion (q) 
0.13 
p*q 
0.1131 
Standard error (S.E.) 
0.03363 
Level of significance 
5% 
Confidence interval 
95% 
Zstatistic (5%) 
1.959964 
Z*SE 
0.065914 
Upper confidence limit 
0.935914 
Lower confidence limit 
0.804086 
A point estimate estimates a parameter by a single number. An interval estimate is an interval of numbers that are probabilistic values for the parameter. The unbiased estimate of sample proportion of number of successes (x) in a sample of size n is given as, phat = (x/n).
95% Confidence Interval
The unbiased point estimate of the population proportion is calculated as 0.87.
The 95% confidence interval for the population proportion of satisfied customers is = (0.804086, 0.935914).
Athens 
Thessaloniki 

Total Count 
50 
Total Count 
50 

Satisfied customer (n1) 
49 
Satisfied customer (n2) 
38 

population proportion (π1) 
0.98 
population proportion (π2) 
0.76 
Zstatistic calculation 

π_{1 } π_{2} 
0.22 
π 
0.87 
π*(1π) 
0.1131 
(1/n_{1}) 
0.02 
(1/n_{2}) 
0.02 
(1/n_{1} + 1/n_{2}) 
0.04 
0.067260687 

Zstatistic 
3.270855684 
Confidence interval 

(π_{1(}1π_{1})/n_{1}) 
0.000392 
(π_{2}(1π_{2})/n_{2}) 
0.003648 
0.063560994 

Level of significance 
5% 
Confidence interval 
95% 
Zstatistic (95%) 
1.959963985 
Confidence interval (95%) 
0.12457726 
Lower confidence limit 
0.09542274 
Upper confidence limit 
0.34457726 
Here, π_{1} denote the population proportion of satisfied customers in Athens and π_{2} the population proportion of satisfied customers in Thessaloniki. The 95% confidence interval estimate of the difference (π_{1 }– π_{2}) is calculated (0.09542274, 0.34457726). It infers that the difference of proportions of satisfied customers in both the cities varies from 0.0954 to 0.3445 with the probability 95%.
We create an additional binary variable (Cost_new) by classifying the average cost of meal per person in Euros as following:
Cost_new: Low when Meal cost per person <= 35 Euros and High when Meal cost per person > 35 Euros.
Low Cost/Meal Cost per Person (<=35) 
Customer satisfaction for Low Cost Meal 
High Cost/Meal Cost per Person (>35) 
Customer satisfaction for High Cost Meal 
25 
3.33 
50 
3.17 
33 
3.33 
38 
3.00 
34 
3.33 
43 
3.17 
35 
3.33 
56 
3.83 
22 
2.83 
51 
3.83 
14 
3.67 
36 
3.83 
27 
3.17 
41 
3.17 
35 
3.17 
44 
3.50 
31 
2.83 
39 
3.50 
34 
3.17 
49 
4.00 
30 
2.33 
37 
3.33 
26 
3.33 
40 
2.83 
35 
3.00 
50 
3.50 
32 
3.00 
50 
3.50 
23 
3.50 
45 
3.50 
31 
3.50 
44 
3.83 
29 
2.83 
38 
2.83 
29 
3.00 
44 
3.17 
27 
3.00 
51 
3.50 
24 
2.33 
44 
3.17 
34 
2.83 
39 
3.50 
23 
2.83 
50 
4.00 
30 
2.33 
48 
3.67 
32 
3.17 
48 
3.67 
25 
2.50 
42 
3.67 
29 
2.00 
63 
4.00 
31 
3.50 
36 
3.50 
26 
2.17 
38 
3.00 
34 
2.83 
53 
3.33 
23 
2.83 
39 
3.17 
32 
2.67 
45 
3.00 
30 
2.83 
37 
3.33 
28 
1.83 
39 
3.17 
33 
2.67 
53 
3.50 
26 
2.00 
37 
3.00 
26 
2.83 
37 
2.33 
24 
1.83 
38 
3.17 
31 
3.50 
37 
3.67 
30 
3.17 
38 
3.00 
30 
3.00 
39 
3.17 
27 
2.67 
36 
3.00 
26 
2.67 
38 
3.33 
28 
2.00 
44 
3.33 
33 
4.00 
44 
3.83 
32 
2.50 
43 
3.17 
25 
2.33 
41 
3.33 
51 
4.17 

48 
3.67 

39 
3.00 

55 
3.33 

38 
3.00 

51 
4.50 

38 
3.33 

38 
3.00 
The hypothesis are
Null hypothesis (H_{0}): The averages of customer satisfaction levels are equal for both the high cost and low cost meals, that is, (μ_{1} = μ_{2}).
Alternative hypothesis (H_{A}): The averages of customer satisfaction levels are unequal for both the high and low cost meals, that is, (μ_{1} ≠ μ_{2}).
ZTest: Two Sample for Means 

Customer satisfaction for Low Cost Meal 
Customer satisfaction for High Cost Meal 

Mean 
2.858405797 
3.388950617 
Known Variance 
0.26 
0.15 
Observations 
46 
54 
Hypothesized Mean Difference 
0 

z 
5.778425733 

P(Z<=z) onetail 
3.77014E09 

z Critical onetail 
2.326347874 

P(Z<=z) twotail 
7.54028E09 

z Critical twotail 
2.575829304 
We executed two samples Ztest for equality of means of customer satisfaction levels at 1% level of significance. The sample sizes are not equal for satisfaction levels of both types of cost prices of meals. Here, variances are also known for both data columns. Therefore, we applied two samples Ztest rather than two samples paired ttest.
Our calculated Zstatistic is (5.778425733). For one tail ztest, the calculated pvalue is 3.77014E09 (0.0). It is less than 0.01 (α=1%). Therefore, the null hypothesis is rejected at 1% level of significance.
Hence, it could be concluded that the assertion of equality of averages of satisfaction levels for both types of meals (low cost and high cost) is false in the 99% confidence interval.
μ_{1 } 
3.35 
μ_{2} 
2.94 
μ_{1}  μ_{2} 
0.41 
(s_{1}^{2})/n_{1} 
0.002411387 
(s_{2}^{2})/n_{2} 
0.006751927 
0.009163314 

standard error 
0.095725202 
Zstatistic 
4.281004267 
Confidence interval 
95% 
Upper confidence limit 
0.597421397 
Lower confidence limit 
0.222178603 
pvalue 
9.30259E06 
The means of average cost of the meal per person in Euros (variable Meal cost per Person) in Athens and in Thessaloniki. The 95% confidence interval of the difference of two means of the average cost of meals in Euros is (0.222178603, 0.597421397). Hence, there is 95% probability of being the difference of average cost of meals in Euros in the interval (0.222178603, 0.597421397).
Conclusion:
From the previous five tasks, it could be concluded that average cost of the meal per person is not equal to €35. The insistence of minister of tourism in Greece about average meal cost per person in Euros is found to be true. The equality of satisfaction level for high and low costs of meals is absent.
References:
Chen, Zhongxue, and Saralees Nadarajah. "On the optimally weighted ztest for combining probabilities from independent studies." Computational Statistics & Data Analysis 70 (2014): 387394.
De Winter, Joost CF. "Using the Student's ttest with extremely small sample sizes." Practical Assessment, Research & Evaluation 18, no. 10 (2013).
Jin, Fulai, Yan Li, Jesse R. Dixon, Siddarth Selvaraj, Zhen Ye, Ah Young Lee, ChiaAn Yen, Anthony D. Schmitt, Celso A. Espinoza, and Bing Ren. "A highresolution map of the threedimensional chromatin interactome in human cells." Nature 503, no. 7475 (2013): 290294.
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