Question

Question: Ambiance of 5-star hotels. Although invisible and intangible, ambient conditions such as air quality , temperature , odor/aroma , music , noise level , and overall image may affect guests’ satisfaction with their stay at a hotel. A study in the Journal of Hospitality Marketing & Management (Vol. 24, 2015) was designed to assess the effect of each of these ambient factors on customer satisfaction with the hotel . Using a survey, researchers collected data for a sample of 422 guests at 5-star hotels. All variables were measured as an average of several 5-point questionnaire responses. The results of the multiple regression are summarized in the table on the next page.

1. Write the equation of a first-order model for hotel image as a function of the six ambient conditions.
2. Give a practical interpretation of each of the b-estimates shown.
3. A 99% confidence interval for is (.350, .576). Give a practical interpretation of this result.
4. Interpret the value of adjusted .
5. Is there sufficient evidence that the overall model is statistically useful for predicting hotel image ? Test using a = .01.

Short answer

(A) The equation for the hotel is$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{6}}\mathbf{+}\mathbf{\epsilon }$

(B) Overall, all the beta values are positive indicating a positive impact on their overall stay at the hotel, however the beta values are very low meaning that the hotel has a lot to improve upon.

(C) 99% confidence interval for ${\mathbf{\beta }}_{\mathbf{6}}$ is (0.350, 0.576) meaning that we are sure to conclude that the value of ${\mathbf{\beta }}_{\mathbf{6}}$ lies within the interval (0.350, 0.576) with 99% accuracy.

(D) The value of adjusted ${\mathbf{R}}^{\mathbf{2}}$ is 0.501 which means that around 50% variation in the variable is explained by the independent variables. The value indicates that the variables are not the best fit to the data.

(E) At least one of the parameters ${\mathbf{\beta }}_{\mathbf{1}},{\mathbf{\beta }}_{\mathbf{2}},...,{\mathbf{\beta }}_6$ is non zero. We have sufficient evidence to conclude that the overall model is not statistically useful to predict hotel image.

Step-by-step solution

Step-by-Step Solution  Step 1: first order model equation

The first order model for the hotel image(Y) as a function of the six ambient conditions can be written as

$\left(\mathbf{y}\right)\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{6}}\mathbf{+}\mathbf{\epsilon }$

Where,

${\mathbf{\beta }}_{\mathbf{1}}\mathbf{,}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{,}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{,}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{,}{\mathbf{\beta }}_{\mathbf{5}}$are the model parameters

$X_1$is air quality

$X_2$is temperature

${\mathbf{x}}_3$is odor/aroma

${\mathbf{x}}_4$is music

${\mathbf{x}}_5$is noise level

${\mathbf{x}}_6$is overall image

Therefore, the equation for the hotel is$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{6}}\mathbf{+}\mathbf{\epsilon }$

Interpretation of beta values

The values for${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{122}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{018}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{124}\mathbf{,}{\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{119}\mathbf{,}{\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{101}$and${\mathbf{x}}_6\mathbf{=}\mathbf{0}.463$which can be interpreted as

For$X_1$, the value 0.122 indicates that since it’s a positive value, the guests are satisfied with air quality however, a lower value indicates that there is a lot to improve in for the hotel.

Forlocalid="1651776167603" ${\mathbf{x}}_2$ , the value 0.018 indicates that since it’s a positive value, the guests are satisfied with temperature however, it’s a very low value indicates that there is a lot to improve in for the hotel.

Forlocalid="1651776182169">x3, the value 0.124 indicates that since it’s a positive value, the guests are satisfied with air odor/aroma around the hotel, however, a lower value indicates that there is a lot to improve in for the hotel.

Forlocalid="1651776192792" ${\mathbf{x}}_4$, the value 0.119 indicates that since it’s a positive value, the guests are satisfied with music however, a lower value indicates that there is a lot to improve in for the hotel.

Forlocalid="1651776110861" ${\mathbf{x}}_5$, the value 0.101 indicates that since it’s a positive value, the guests are satisfied with air noise level however, a lower value indicates that there is a lot to improve in for the hotel. The hotel can promote noise reducing events and activities around the hotel.

Forlocalid="1651776097046" ${\mathbf{x}}_{\mathbf{6}}$, the value 0.463 indicates that since it’s a positive value, the guests are satisfied with their overall stay at the hotel, a lower value indicates that there is a lot to improve in for the hotel.

Overall, all the beta values are positive indicating a positive impact on their overall stay at the hotel, however the beta values are very low meaning that the hotel has a lot to improve upon.

Confidence interval for  β6

99% confidence interval for ${\mathbf{\beta }}_{\mathbf{6}}$ is (0.350, 0.576) meaning that we are sure to conclude that the value of ${\mathbf{\beta }}_{\mathbf{6}}$ lies within the interval (0.350, 0.576) with 99% accuracy. Even the value of ${\mathbf{x}}_6$ estimate calculated after the research is 0.463 which lies within the interval.

Step 4: R2and adjusted R2

The value of ${\mathbf{R}}^{\mathbf{2}}$ is 0.508 which means that around 50.8% of the variation in the variables is explained by the model. The value of 50.8% is not very high indicating that the model might not be a good fit for the data.

The value of adjusted ${\mathbf{R}}^{\mathbf{2}}$ is 0.501 which means that around 50% variation in the variable is explained by the independent variables. The value indicates that the variables are not the best fit to the data.

Overall significance of the model