How to calculate confidence for non standard bands?

So I have a dataset : dt<-c(1106,1402, 827, 781,876, 1134,1014, 964, 848, 814, 772, 912, 923, 996, 569, 774, 1389, 900) lets assume a normal curve and running t.test(dt) I get

One Sample t-test

data: dt t = 19.057, df = 17, p-value = 6.579e-13 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: 839.9344 1049.0656 sample estimates: mean of x 944.5

Which is all standard fare for 95% confidence interval.

But what I'd like to find is what is the confidence in a specific range like from 850 to 900. Because I want to know the probability that the next datapoint will fall between 850 and 900. Which package::function can do this?

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I don't know of a built-in function but it's not too too hard to compute using pt() , the cumulative distribution function for Student's t:

dd <- c(1106,1402, 827, 781,876, 1134,1014, 
        964, 848, 814, 772, 912, 923, 996, 569, 774, 1389, 900)
m <- mean(dd)
s <- sd(dd)

Now we (1) convert the desired range to "t statistic" scale (subtract mean and divide by sd) and (2) compute the cumulative probability of x<lower_bound and x<upper_bound

probs <- pt((c(850,900)-m)/s,df=length(dd)-1)

The probability of a value falling in the range is the difference of these two values.

diff(probs)  ## 0.08805229

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