What is the best training programme to improve 1.5 mile run times? For reference, my starting point is about 9:30, aiming for 8:30.
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At that short a distance I'd say run more. First and foremost, you need your body to get acustomed to running faster.
So doing sessions of 400m and 800m sprints - about 0.25mile and 0.5mile respectively, aiming to get towards a pace at or below your goal pace will help.
Additionally, keeping consistant records and lot of practise will get you there quickly.
A plan like that below (with adequate rest) should help get you towards the speed you want:
On every day that you run, go for at least 30 minutes. When you do intervals, go for consistent pacing, not speed--quick but relaxed. Alternate between workout days and easy running/jogging days. Don't bother too much with speed work right now, for 1.5 miles you need to develop aerobic fitness more than anything else. If you haven't been training much/at all, you should probably just do easy runs of 30+ minutes each for several weeks before adding intervals. A typical mid to long distance (>= 1500m races, often down to 800m) training program will start with "base training" (easy runs only), then progress into alternating easy days and longer, endurance-oriented workouts, then add some race-pace and faster intervals. 600m-1000m is where "sprinters" and "endurance runners" overlap (in a grossly over-simplified 2-party model), so if you're going for 1.5 miles I would recommend approaching it as a distance runner rather than a sprinter.
For intervals, you might try starting 3-4x800m in 3:20-3:30 with 60-90 seconds rest. I got that pace range both by estimating relative to your race pace and by using this pace calculator. McMillan's "steady state" pace would be good for some 2-3 mile runs to build aerobic endurance as well. Doing a few shorter, faster workouts will probably help you, too, but it's not what's most important for a 1.5 mile race.
Finally, be sure to build up your mileage and intensity gradually to avoid injury.