find the 90% confidence interval for the variance and standard deviation of a sample of 16 and a standard deviation of 2.1 Assume the variable is normally distributed.

There are different ways you can do this kind of problem, but the formula below might be one of the easier ways:

s/[1 + (1.645/√2n)]
..to..
s/[1 - (1.645/√2n)]

...where s = standard deviation, 1.645 represents the 90% confidence interval using a z-table, and n = sample size.

Substituting your values:
(2.1)/[1 + (1.645/√2*16)]
..to..
(2.1)/[1 - (1.645/√2*16)]

Note: After you finish the above calculations, square the standard deviation values to find the variance values.

I hope this will help.

To find the 90% confidence interval for the variance and standard deviation, we need to use the Chi-Square distribution.

First, let's calculate the degrees of freedom for the Chi-Square distribution. The sample size is 16, so the degrees of freedom (df) will be (n - 1), where n is the sample size. In this case, df = 16 - 1 = 15.

Next, we need to find the critical values for the Chi-Square distribution corresponding to a 90% confidence level and the degrees of freedom. The critical values determine the range within which the true variance or standard deviation is likely to be.

We look up the critical values in the Chi-Square table or use statistical software. The lower critical value (χ²L) is the value below which 5% of the distribution lies, and the upper critical value (χ²U) is the value below which 95% of the distribution lies for a two-tailed test.

For a 90% confidence level and 15 degrees of freedom, the lower critical value is χ²L = 6.262 and the upper critical value is χ²U = 24.996.

Now we can calculate the confidence interval.

1. Confidence interval for the variance:
Lower Limit = [(n-1)*S²] / χ²U
Upper Limit = [(n-1)*S²] / χ²L

Plugging the values:
Lower Limit = 15*2.1² / 24.996 ≈ 3.155
Upper Limit = 15*2.1² / 6.262 ≈ 10.616

Therefore, the 90% confidence interval for the variance is (3.155, 10.616).

2. Confidence interval for the standard deviation:
To find the confidence interval for the standard deviation, we take the square root of the confidence interval for the variance.

Lower Limit = √(Lower Limit of variance)
Upper Limit = √(Upper Limit of variance)

Plugging the values:
Lower Limit = √(3.155) ≈ 1.776
Upper Limit = √(10.616) ≈ 3.259

Therefore, the 90% confidence interval for the standard deviation is (1.776, 3.259).