What is a confidence interval in Statistics?

Question

Please explain, using an example, what a confidence interval is and how we would apply it to sample scenario.

Answer 1

A confidence interval is a type of inferential statistic - that is, an inferred measure meant to describe of whole of a group that is based on a sample of that whole group. The crux of inferential statistics is to be able to say something about all of something, but since we cannot realistically measure all, we take a sample of the whole group that is meant to be representative, from which we generalize its statistics to the larger population.

A confidence interval is a range of values with a lower bound and an upper bound that is meant to describe with a certain level of confidence, the likelihood, the range wherein we expect to find the true population parameter. A typical confidence interval includes a point estimate value, this is our 'best guess' at the population parameter, which is computed from sample data (a common measure that we make confidence intervals for are means, so the sample mean would be the point estimate which represents our 'best guess' at the population parameter). Then, subtracting and adding a margin of error amount (E; dependent on a chose level of significance) onto the point estimate to generate the lower and upper bound values for the interval. A general pseudo-formula for a confidence interval (CI) —looks like the following:

CI = Pt. Estimate ± E

Diagram illustrating a typical CI:

E E

[—————————•—————————]

lower bound Point upper bound

Estimate

EXAMPLE

A weather forecaster called for significant snowfall accumulation for a winter storm. The city was blanketed by a hefty amount of snow and the forecaster wishes to describe the snowfall that fell in the city. He travels around the city taking snow depth measurements at n = 45 different locations that will serve as a representative collection of sites (thus snow depths) across the city. The mean snow depth, x̄, is 9 inches and serves as the 'best guess' or the point estimate of the true mean snowfall that fell across the entire city (imagine if you were to measure every infinitesimally small location across the whole city then computed their average, this is the population parameter for which we are crafting our interval for). The forecaster also computes the standard deviation for the 45 different snow depths, 2 inches, and they also decide that they are making a 95% CI, therefore within the margin of error, the t value they will use is 2.015. The following formulas show how the forecaster writes the CI out symbolically and then how he inputs his data into the formula to compute the lower and the upper bound of the 95% CI.

CI (95%) = x̄ ± t * ( s / √n )

—————

↑

this is the margin of error, E

CI (95%) = 9 ± 2.015 * ( 2 / √ 45 )

= 9 ± 0.6009

Therefore, the weather forecaster finds that he is 95% confident that the true mean snowfall that fell across the city in the recent snowstorm is approximately:

CI (95%) = (8.40 inches, 9.60 inches)

Answer 2

A confidence interval (CI) in statistics is a range of values, derived from sample data, that is likely to contain the true population parameter (such as the mean, proportion, or difference between means) with a certain level of confidence. It provides an estimate of the uncertainty associated with a sample statistic.

Key Components:

1. Point Estimate: The best single estimate of the population parameter (e.g., sample mean or proportion).
2. Margin of Error: The range around the point estimate, based on the desired level of confidence (e.g., 95%) and variability in the data.
3. Confidence Level: The probability that the interval includes the true population parameter, commonly set at 90%, 95%, or 99%.

Example Scenario:

Imagine we are studying the average height of students in a school. We take a random sample of 50 students and measure their heights. The sample mean height is 160 cm, and the sample standard deviation is 10 cm.

We want to calculate a 95% confidence interval for the true mean height of all students in the school.

Steps to Calculate the Confidence Interval:

1.Identify the Point Estimate (Sample Mean):

x-bar = 160 cm.

2.Determine the Confidence Level and Find the Z-Score:

For a 95% confidence level, the critical Z-score is approximately 1.96.

3.Calculate the Standard Error (SE):

The standard error is calculated as:

SE=Standard Deviation/Sample Size=1050/ SQrt (50) ≈1.41

4.Calculate the Margin of Error (ME):

ME=Z×SE=1.96×1.41≈2.76

5.Construct the Confidence Interval:

The confidence interval is:

CI=xˉ±ME=160±2.76= (157.24, 162.76).

Interpretation:

We are 95% confident that the true mean height of all students in the school lies between 157.24 cm and 162.76 cm. This does not mean that 95% of the students' heights fall within this range; rather, if we were to take many random samples and calculate a confidence interval for each, about 95% of those intervals would contain the true population mean.

Application:

Confidence intervals are widely used in fields like healthcare, business, and social sciences to make inferences about populations from sample data. For instance, a pharmaceutical company might use a confidence interval to estimate the average effectiveness of a new drug, allowing them to assess its reliability and make informed decisions. I hope this helps. If you have any additional questions, please let me know. Take care,

Dr. Christal-Joy Turner

Answer 3

A confidence interval is used to showcase how likely an outcome or score seen in a sample is likely to happen in the population. In other words, the confidence interval shows the range of values of which we are, say, 95% confident that the true measure in the population lies given our sample statistics. This interval would be our confidence interval at 95% confidence. Please see the video above for a breakdown of confidence intervals with an example.