Confidence Interval: Basic Statistics Lecture Series Lecture #11

You'll remember last time, I covered hypothesis testing of proportions and the time before that, hypothesis testing of a sample with a mean and standard deviation.  This time, I'll cover the concept of confidence intervals.

Confidence intervals are of the form μ_1-α ∈ (a, b)_1-α, where a and b are two numbers such that a<b, α is the significance level as covered in hypothesis testing, and μ is the actual population mean (not the sample mean). This is a the statement of there being a [(1-α)*100]% probability that the true population mean will be somewhere between a and b.  The obvious question is "How do we find a and b?".  Here, I will describe the process.

Step 1. Find the Fundamental Statistics

The first thing we need to find the fundamental statistics, the mean, standard deviation, and the sample size.  The sample mean is typically referred to as the point estimate by most statistics text books.  This is because the point estimate of the population mean is the sample mean.  These will eventually be vital to finding the interval.  It is also important to find the z-value for the particular confidence level 1-α for which we are interested in finding the confidence interval.  Remember from the hypothesis testing entries that the z-value (without use of calculus or statistics software) comes from the z-table.  Since this is an interval where there will be two tails, we need to find the z-values for 1-^α⁄₂ ​.  Here is a table of z-values for common confidence levels:

Confidence Level	z
0.7	1.04
0.75	1.15
0.8	1.28
0.85	1.44
0.9	1.645
0.92	1.75
0.95	1.96
0.96	2.05
0.98	2.33
0.99	2.58

Step 2. Calculating the Standard Error and Margin of Error

The next step is to calculate the standard error of the sample.  For the case where we have the sampel mean and standard deviation, it is simple $\sigma_{E}=\frac{s}{\sqrt{n}}$, and for the proportion case, it is $\sigma_{E}=\sqrt{\frac{p*(1-P)}{n}}$.  This is the deviation of the data points per data point.  In order to go from the standard error to the margin of error, we simply multiply the standard error by the z-value for the confidence level, $\sigma_{M}=z*\sigma_{E}$.

Step 3. Getting the Confidence Interval

This is simple enough of a calculation to perform.  You'll notice that in the second paragraph that I displayed the confidence interval as (a, b)_1-α.  The value for a is subtracting the margin of error from the mean (point estimate), $a=\overline{x}-\sigma_{M}$, where b is taking the sample mean and to it, add the margin of error, $b=\overline{x}+\sigma_{M}$.  So the confidence interval is given by: $$\mu_{1-\alpha} \in (\overline{x}-\sigma_{M}, \overline{x}+\sigma_{M})$$This means that there is a [(1-α)*100]% probability that the true population mean falls between $\overline{x}-\sigma_{M}$ and $\overline{x}+\sigma_{M}$.

You'll notice that the point estimate is a single value and not a confidence interval.  This stems from the notion that the value of α approaches 1 as our confidence approaches 0%.  This means that the z-value approaches 0 as α approaches 1.  This also means that the value of the Margin of Error approaches 0, since it is the z value times a value.  At 0% confidence, z=0, so the margin of error is zero, since zero times anything is zero.  This means that the population mean is "between" $\overline{x}-$ and $\overline{x}+0$, which means that the population mean at a 0% confidence is the population mean, which is the statement of the point estimate.

That's it for confidence interval.  If you have any questions, please leave a comment.  Next time, I will be covering types of statistical error.  Until then, stay curious.

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K. "Alan" Eister has his bacholers of Science in Chemistry. He is also a tutor for Varsity Tutors.  If you feel you need any further help on any of the topics covered, you can sign up for tutoring sessions here.