Logistic Regression with Only Negative Predictions

Course Q&A Machine Learning Specialization Supervised ML: Regression and Classification
1

If all possible values of z (predictions from linear regression) are negative, then if we use a threshold of 0.5, wouldn’t logistic regression always predict y=0?

I’m assuming in this case we would have to change the threshold. I’m just unsure because Andrew only ever mentions a threshold of 0.5 in the video and lab and also gave this as a rule:

As you can see,  𝑔(𝑧)>=0.5  for  𝑧>=0 
For a logistic regression model,  𝑧=𝐰⋅𝐱+𝑏 . Therefore,

if  𝐰⋅𝐱+𝑏>=0 , the model predicts  𝑦=1 
if  𝐰⋅𝐱+𝑏<0 , the model predicts  𝑦=0

Any clarification would be appreciated

2

If you’re including the sigmoid function, then the threshold is >= 0.5
If you’re not including the sigmoid function, then the threshold is >= 0.0.

That’s because sigmoid(0) = 0.5.