Explanation:
To find a pair of values for the sample standard deviation such that the probability it lies between them is 0.90, we need to use the chi-square distribution. The formula for the chi-square distribution with (n-1) degrees of freedom for the sample standard deviation, where 'n' is the sample size, is:
chi^2 = ((n-1) * s^2) / (sigma^2)
Where:
chi^2 is the chi-square statistic.
n is the sample size (20 in this case).
s is the sample standard deviation.
sigma is the population standard deviation (given as 2% or 0.02).
We want to find two values of s such that the probability that chi^2 falls between them is 0.90. This means we need to find the 5th and 95th percentiles of the chi-square distribution with 19 degrees of freedom because (20-1=19).
You can use a chi-square table or calculator to find these percentiles. For a chi-square distribution with 19 degrees of freedom, the 5th percentile is approximately 9.235 and the 95th percentile is approximately 32.852.
Now, set up two equations using the chi-square formula:
For the 5th percentile:
9.235 = ((20-1) * s1^2) / (0.02^2)
For the 95th percentile:
32.852 = ((20-1) * s2^2) / (0.02^2)
Solve these equations for s1 and s2:
For s1:
s1^2 = (9.235 * (0.02^2)) / (20-1)
s1 ≈ 0.01356
For s2:
s2^2 = (32.852 * (0.02^2)) / (20-1)
s2 ≈ 0.02541
So, any pair of values such that the probability the sample standard deviation lies between them is 0.90 is approximately s1 ≈ 0.01356 and s2 ≈ 0.02541.