Question

Expressing Confidence Intervals Example 2 showed how the statistics ofn= 22 ands= 14.3 result in this 95% confidence interval estimate of $\sigma$: 11.0 < $\sigma$ < 20.4. That confidence interval can also be expressed as (11.0, 20.4), but it cannot be expressed as 15.7$\pm$ 4.7. Given that 15.7$\pm$4.7 results in values of 11.0 and 20.4, why is it wrong to express the confidence interval as 15.7$\pm$4.7?

Short answer

It is wrong to express the confidence interval estimate of $\sigma$equal to (11.0, 20.4) as $\left(15.7\pm 4.7\right)$because of the following two reasons:

• The confidence interval of $\sigma$is not symmetric about the sample standard deviation and cannot be expressed as $\left(s\pm E\right)$.
• It can be believed that s takes the value equal to 15.7 which is incorrect.

Step-by-step solution

Given information

The size of the sample is $n=22$.

The sample standard deviation is $s=14.3$.

The 95% confidence interval estimate of $\sigma$is $11.0<\sigma <20.4$.

Correct expression of the confidence interval

The confidence interval to estimate the population standard deviation$\left(\sigma \right)$ has the following expression:

$CI=\sqrt{\frac{\left(n-1\right)s^2}{{\chi }_R^2}}<\sigma <\sqrt{\frac{\left(n-1\right)s^2}{{\chi }_L^2}}$

where

n is the sample size

s is the sample standard deviation

${\chi }_R^2$is the right-tailed critical value of ${\chi }^2$with degrees of freedom$\left(n-1\right)$

${\chi }_L^2$is the left-tailed critical value of ${\chi }^2$with degrees of freedom$\left(n-1\right)$

Here, the value of the confidence interval is equal to (11.0, 20.4).

It is incorrect to express the confidence interval as role="math" localid="1648040974197" $\left(15.7\pm 4.7\right)$because the confidence interval of role="math" localid="1648040987212" $\sigma$ is not symmetric about the sample standard deviation.

Also, it makes the reader believe that the value of the sample standard deviation is equal to 15.7 by comparing the value as follows;

$\left(s\pm \text{Margin}\text{of}\text{Error}\right)=\left(15.7\pm 4.7\right)$

It can be assumed that s takes the value equal to 15.7 and the margin of error takes a value equal to 4.7 when in reality, the value of the sample standard deviation is equal to 14.3.

Therefore, $\left(15.7\pm 4.7\right)$is an absolutely incorrect way of expressing the confidence interval.