Question $3.$ Evaluating Staff Productivity with Two Management Styles:

An HR manager wants to determine if two management styles result in different levels of staff productivity. Management Style $1$ is the current approach, and Style $2$ is a new approach. A test with $15$ employees for each style is conducted, with productivity scores recorded. The standard deviation for Style $1$ is $5$ points, and for Style $2$ it is $6$ points. The sample means for Style $1$ and Style $2$ are $75$ and $78$ points, respectively. Using a significance level of $0\mathrm{.}05,$ is there a significant difference in the mean productivity scores?