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Question 1. Suppose that you run four restaurants. Let A be 0 and 1 for am and $...

Apr 11, 2024

Question 1. Suppose that you run four restaurants. Let A be 0 and 1 for am and

\mathrm{pm}

. Let

B_{1}, B_{2}, B_{3}

for dummy variables corresponding to restaurants labelled 1,2 , and 3 respectively. Let

\mathrm{Y}

be 0 if no complaint is received and 1 otherwise. (a) [10 marks

]

Consider the logistic regression model

\begin{array}{l} \operatorname{logit}[P(Y=1)]= \\ -0.2-0.9 A-1.3 B_{1}-0.6 B_{2}-1.2 B_{3}+1.3 A B_{1}+0.7 A B_{2}+0.3 A B_{3} \end{array}

Find the probability

P(Y=1)

for each of the following cases. \begin{tabular}{|l|l|l|l|l|} \hline

\mathrm{P}(\mathrm{Y}=1)

& Restaurant 1 & Restaurant 2 & Restaurant 3 & Restaurant 4 \\ \hline

\mathrm{am}

& & & & \\ \hline

\mathrm{pm}

& & & & \\ \hline \end{tabular}

Solution by Steps

step 1

To calculate the probability

P(Y=1)

for each case, we first need to understand the logistic regression model given. The logit function is defined as

\operatorname{logit}[P(Y=1)] = \log\left(\frac{P(Y=1)}{1-P(Y=1)}\right)

step 2

For Restaurant 1 in the AM, we have

A=0

B_1=1

B_2=0

, and

B_3=0

. Plugging these values into the logistic regression model, we get:

\operatorname{logit}[P(Y=1)] = -0.2 - 0.9(0) - 1.3(1) - 0.6(0) - 1.2(0) + 1.3(0)(1) + 0.7(0)(0) + 0.3(0)(0) = -1.5.

step 3

To find

P(Y=1)

, we solve for it using the inverse logit function:

P(Y=1) = \frac{e^{\operatorname{logit}[P(Y=1)]}}{1 + e^{\operatorname{logit}[P(Y=1)]}} = \frac{e^{-1.5}}{1 + e^{-1.5}}.

step 4

Repeat steps 2 and 3 for each combination of AM/PM and Restaurant 1/2/3/4. Note that for Restaurant 4, since it is not included in the model, we assume

B_1=B_2=B_3=0

step 5

Calculate the probabilities for each case using the inverse logit function as shown in step 3

Answer

For Restaurant 1 in the AM:

P(Y=1) = \frac{e^{-1.5}}{1 + e^{-1.5}}

For Restaurant 2 in the AM:

P(Y=1) = \frac{e^{-0.8}}{1 + e^{-0.8}}

For Restaurant 3 in the AM:

P(Y=1) = \frac{e^{-1.4}}{1 + e^{-1.4}}

For Restaurant 4 in the AM:

P(Y=1) = \frac{e^{-0.2}}{1 + e^{-0.2}}

For Restaurant 1 in the PM:

P(Y=1) = \frac{e^{-0.6}}{1 + e^{-0.6}}

For Restaurant 2 in the PM:

P(Y=1) = \frac{e^{0.1}}{1 + e^{0.1}}

For Restaurant 3 in the PM:

P(Y=1) = \frac{e^{-0.6}}{1 + e^{-0.6}}

For Restaurant 4 in the PM:

P(Y=1) = \frac{e^{0.7}}{1 + e^{0.7}}

Key Concept

The key concept is to use the logistic regression model to calculate the logit of the probability and then apply the inverse logit function to find the actual probability.

Explanation

The logistic regression model provides the logit of the probability of an event. By substituting the appropriate values for AM/PM and the restaurant dummy variables, we can calculate the logit for each case. The inverse logit function is then used to convert the logit back to a probability.