Multiple testing models have become an important part of statistical applications. Typically they can be presented as having M hypotheses each of which concerns an individual parameter. In addition to testing each of these hypotheses, there is often a desire to obtain interval estimates for the parameters. The use of stepwise procedures arises because single-step procedures are extremely conservative. Unfortunately research into the construction of useful, computationally feasible interval estimates corresponding to stepwise procedures has been slow. We present an alternative method of constructing multiple testing procedures (MTPs) that easily admits corresponding interval estimates. The new method places greater focus on each hypothesis separately while still using all the data. This method is particularly effective in the dependent case. Not only do these new MTPs perform as well as commonly used stepwise procedures but they also have a practical interval property not usually shared by stepwise procedures. That is, acceptance regions have desirable convexity properties. Furthermore, interval estimates associated with these tests are easily obtained. In addition, these intervals (i) are typically shorter than those based on the Bonferroni, Scheff, Tukey or Dunnett method when they are applicable, (ii) are less likely to contain the null point falsely than other methods do, (iii) are informative, i.e. they are all finite in the two-sided case, unlike some constructed by other methods which often are infinite, (iv) have a form of the interval property.