8.7:
Estimating Population Standard Deviation
Sample statistics are often used to estimate population parameters. For instance, the sample standard deviation can reasonably estimate the population standard deviation if n is sufficiently large.
But, the sample standard deviation often underestimates or overestimates the population standard deviation. So, confidence intervals are determined to compensate for these biases.
It's important to note that only randomly selected samples from a normally distributed population can be used to estimate population parameters.
Consider the data on temperature variations with a sample standard deviation of 1.5 degrees Celsius. Using this, one can estimate the population standard deviation with a suitable confidence interval, say 95%.
First, using the chi-square table, find the right and left critical values. Then determine the confidence intervals of the population variance, separately for the left and the right critical values.
Taking the square root of these values gives the confidence intervals of the population standard deviation, which can be rounded off for convenience.
So, one can say with 95% confidence that the true value of the population standard deviation lies between 1.03 and 2.74 degrees Celsius.
8.7:
Estimating Population Standard Deviation
When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and standard deviation, one can construct confidence intervals for the population standard deviations at a suitable significance level, such as 95%. A confidence interval is an interval of numbers. It provides a range of reasonable values in which we expect the population parameter to fall. There is no guarantee that a given confidence interval does capture the population standard deviation, but there is a predictable probability of success. The critical values in the right and left tails of the distribution curve provide the confidence intervals of the population standard deviation.
This text is adapted from Openstax, Introductory Statistics, Section 8, Confidence Interval