To test this I derived a formula to calculate the coefficient of friction (top left). The only variable I needed to complete the calculations was the angle at which the shoe started slipping so I set up a way to gather this info (shown on left). When each shoe started slipping I would record the angle of the board shown on my phone. I then created a data table with the angle and the corresponding coefficient of friction
These are the results of my experiment (right). Overall, while climbing shoes seemed to have a slightly larger coefficient of friction relative to regular shoes, the difference is not as great as I would have expected. There were some flaws in my experiment design. The climbing shoes I tested were a few sizes smaller than the regular shoes I tested and climbing shoes are also extremely arched and intended to have weight on them which could have altered the results.