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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
per person

 One Sample Z-test for estimating confidence interval

Sum

3674

Mean (μ)

36.74

Unbiased estimator of mean (X-bar)

37.11111111

Standard deviation (s)

9.203227365

(X-bar - μ)

0.371111111

Count (n)

100

Square-root of n

10

s/squareroot(n)

0.920322737

Confidence interval

95%

Level of significance

5%

Z-statistic

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
per person

One sample t-test (one-tail)

X-bar

36.74

Hypothesized mean (μ)

35

(X-bar - μ)

1.74

Standard deviation (s)

9.203227365

Count (n)

100

Degrees of freedom (d.f.)

99

Square-root of n

10

s/squareroot(n)

0.920322737

t-statistics

1.89064111

p-value

0.030798262

Null Hypothesis

Meal cost per person is equal to €35

Decision making

Null Hypothesis rejected

Null Hypothesis (H0): Average Meal cost per person is equal to €35, that is, (μ = 35).

Alternative Hypothesis (HA): Average Meal cost per person is unequal to €35, that is, (μ ≠ 35).

The t-statistic (1.89064111) is calculated as:

t = (sample mean – hypothesized mean)/ SE mean.

The calculated p-value 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 p-value less than 0.05 indicate that mean cost per person in Euros is not normally distributed.

Meal cost
per person

One sample Z-test for estimating confidence interval

Sum

3674

Mean (μ)

36.74

Unbiased estimator of mean (X-bar)

37.11111111

Standard deviation (s)

9.203227365

Count (n)

100

Square-root of n

10

s/squareroot(n)

0.920322737

Confidence interval

90%

Level of significance

10%

Z-statistic

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
per person

One sample t-test (one-tail)

X-bar

36.74

Hypothesized mean (μ)

35

(X-bar - μ)

1.74

Standard deviation (s)

9.203227365

Count (n)

100

Degrees of freedom (d.f.)

99

Square-root of n

10

s/squareroot(n)

0.920322737

t-statistics

1.89064111

p-value

0.030798262

Null Hypothesis

Meal cost per person is equal to €35

Decision making

Null Hypothesis rejected

The calculated p-value 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 (H0): The average meal cost per person is greater than or equals to €40, that is, (μ ≥ 40).

Alternative hypothesis (HA): 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
per person

One sample t-test (one-tail)

X-bar

36.74

Hypothesized mean (μ)

40

(X-bar - μ)

-3.26

Standard deviation (s)

9.203227365

Count (n)

100

Degrees of freedom (d.f.)

99

Square-root of n

10

s/squareroot(n)

0.920322737

t-statistics

-3.542235643

p-value

0.00030344

Null Hypothesis

Meal cost per person is less than €40

Decision making

Null Hypothesis rejected

 Based on outputs one-sample t-test 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
Satisfaction (<=2.5)

Satisfied/Overall Customer  (>2.5)
Satisfaction

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%

Z-statistic (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, p-hat = (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

Z-statistic calculation

π1 - π2

0.22

π

0.87

π*(1-π)

0.1131

(1/n1)

0.02

(1/n2)

0.02

(1/n1  +  1/n2)

0.04

0.067260687

Z-statistic

3.270855684

Confidence interval

1(1-π1)/n1)

0.000392

2(1-π2)/n2)

0.003648

0.063560994

Level of significance

5%

Confidence interval

95%

Z-statistic (95%)

1.959963985

Confidence interval (95%)

0.12457726

Lower confidence limit

0.09542274

Upper confidence limit

0.34457726

Here, π1 denote the population proportion of satisfied customers in Athens and π2 the population proportion of satisfied customers in Thessaloniki. The 95% confidence interval estimate of the difference (π1 – π2) 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 (H0): The averages of customer satisfaction levels are equal for both the high cost and low cost meals, that is, (μ1 = μ2).

Alternative hypothesis (HA): The averages of customer satisfaction levels are unequal for both the high and low cost meals, that is, (μ1 ≠ μ2).

Z-Test: 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) one-tail

3.77014E-09

z Critical one-tail

2.326347874

P(Z<=z) two-tail

7.54028E-09

z Critical two-tail

2.575829304

We executed two samples Z-test 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 Z-test rather than two samples paired t-test.

Our calculated Z-statistic is (-5.778425733). For one tail z-test, the calculated p-value is 3.77014E-09 (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.

μ1

3.35

μ2

2.94

μ1 - μ2

0.41

(s12)/n1

0.002411387

(s22)/n2

0.006751927

0.009163314

standard error

0.095725202

Z-statistic

4.281004267

Confidence interval

95%

Upper confidence limit

0.597421397

Lower confidence limit

0.222178603

p-value

9.30259E-06

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 z-test for combining probabilities from independent studies." Computational Statistics & Data Analysis 70 (2014): 387-394.

De Winter, Joost CF. "Using the Student's t-test 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, Chia-An Yen, Anthony D. Schmitt, Celso A. Espinoza, and Bing Ren. "A high-resolution map of the three-dimensional chromatin interactome in human cells." Nature 503, no. 7475 (2013): 290-294.

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