Binomial Distributions and Confidence Intervals

If I have a binomial distribution, can I work out the confidence interval simply by summing up the probabilities of each set of outcomes until I reach 95%?

e.g. n = 70, p(success) = 0.25. I know that 95% of the distribution falls between 11 and 24.

Can I take this range as the 95% confidence interval or do I need to approximate to a normal distribution and do a z-test? The interval I have selected is evenly distributed +/-6.5 of the mean.

The mean of the data is 17.5, SD is 3.62 using this site

Binomial Distribution Calculator

(and if you put "Between 11 and 24" you will see that the probability is 0.9480 ~= 95%)

Here is a sample of my data if you wish to do some rough calculations

0 0.0000

1 0.0000

2 0.0000

3 0.0000

4 0.0000

5 0.0001

6 0.0003

7 0.0010

8 0.0026

9 0.0059

10 0.0121

*11 0.0219

*12 0.0360

*13 0.0535

*14 0.0726

*15 0.0903

*16 0.1035

*17 0.1096

*18 0.1075

*19 0.0981

*20 0.0834

*21 0.0662

*22 0.0491

*23 0.0342

*24 0.0223

25 0.0137

26 0.0079

27 0.0043

28 0.0022

29 0.0011

30 0.0005

31 0.0002

32 0.0001

33 0.0000

The thing here is I'm using a discrete variable so I don't think it would be right to approximate this to a normal distribution and then do a z-test.

Thanks.

PS: I think I might have posted this in the wrong forum: this might be more college/university level. Feel free to move it over - moderators.