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.

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

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.

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.

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.