Statistics for Library and Information Services by Friedman Alon;

Author:Friedman, Alon; [Friedman, Alon]
Language: eng
Format: epub
Publisher: Rowman & Littlefield Unlimited Model
Published: 2015-08-15T00:00:00+00:00

Figure 10.2. Illustration of the point estimate’s relationship to the confidence interval.

How much uncertainty is associated with a point estimate of a population parameter versus a confidence interval?

Example:

Suppose we want to know what percentage of the public library members in Tampa, Florida, browsed the latest novel from J. K. Rowling, the author of the Harry Potter series? To estimate this, we take a random sample of 100 library members and find out what percentage of those 100 members read J. K. Rowling’s last novel. Then, we say that the sample percentage should come close to the actual percentage of all Tampa public library members who read J. K. Rowling.

In this example:

Population of Interest: All public members of the public library in Tampa

Sample Size: n = 100

Statistic from the sample: The percentage of library members sampled who read J. K. Rowling’s latest novel

Parameter in the Population: The percentage of all Tampa library members who read J. K. Rowling’s latest novel

How do we actually estimate the parameter, though?

10.3 General Estimation Framework

Suppose we want to estimate a parameter (e.g., population proportion, population average, etc.). The first thing to notice is that it would be impossible to exactly pinpoint the value with 100% accuracy without sampling every single member of the population, since there would always be some uncertainty. As a result, the best we can do is to make a guess at the true value, and then include a margin of error based on a certain level of confidence we have in our results. The estimate and the margin of error form something called a confidence interval. A confidence interval is made of two different parts:

1.The point estimate is the sample statistic (this is our best guess at the true parameter value given our sample).

2.The margin of error is added and subtracted from the point estimate to make the interval.

It can also be subdivided into two parts:

a.A critical value from a distribution (more to come on this later)

b.The standard error of the point estimate (more to come on this as well)

The confidence interval (CI) has this form:

CI = (Point Estimate) ± (Margin of Error)

CI = (Point Estimate) ± (Critical Value)*(Standard Error)

Of course there is no guarantee that the true population parameter will be in this interval, so we have to make some sort of statement about the chances that this will be true.

The confidence level is the probability that the interval actually covers the true population parameter. Often, the confidence level is denoted (1 – α), where α is the chance that it does not cover the true parameter. For example, if α = 0.05, then the confidence level is 0.95, or 95%. Thus we would say that we are 95% confident that the interval covers the parameter.

An example of a 95% confidence interval is shown below:

72.85 < μ < 107.15

There is good reason to believe that the population’s mean lies between these two, 72.85 and 107.15, since 95% of the time confidence intervals contain the true mean. If repeated samples

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