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 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 |
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 |
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 |
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 |
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 |
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% |
|
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 |
|
|
(π1(1-π1)/n1) |
0.000392 |
|
(π2(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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