You could, in fact, test whether or not removing the keyword from the page has an effect on it's rankings, but most likely that won't necessarily prove anything. In fact, you may even be more confused, as a page can rank well even if the page doesn't contain the keyword or keyword phrase.
There may be a lot more factoring into these search engine rankings, such as links pointing to the page or the number of social media shares. It could also be related to how often you update certain pages on your site and if you update those pages or not. Another factor could be user engagement on the page, such as how long they spend on the page, bounce rate, or whether or not there are comments (if it's a blog post).
If there's one page that you want to rank (because it happens to convert better for you), then I would focus on link building and social shares.