Question

The following data is for a simple random sample taken from a normal population: 2.41 2.45 2.21 2.32 2.25 2.38

(a) Find a 95% prediction interval.
(b) Find a tolerance interval that contains 90% of the data with 95% confidence

Answer 1

Given, the following data is a simple random sample of size, n = 6, from a normal population,

2.41,2.45,2.21,2.32,2.25,2.38

Part a : Prediction Interval

The probability of the the observation falling in a given interval is then:

where T_a is the upper 100(1 − p/2)^th percentile of Student's t-distribution with n − 1 degrees of freedom and is the sample standard deviation. Therefore, the numbers

are the endpoints of a 100p % prediction interval for .

Here, n = 6 ; = 2.337 ; = 2.571 ; = 0.093 ; n =6 ; p = 0.05

The prediction interval is thus obtained as,

[2.076,2.597]

Part b : Tolerance Interval

The tolerance interval for a data is obtained as,

Using the following R code, we can calculate the above tolerance interval

normtol.int(x = obs, alpha = 0.05, P = 0.90, side = 2)